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Quadratic Formula Calculator
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This online calculator is a quadratic equation solver that will solve a second-order polynomial equation such as ax 2 + bx + c = 0 for x, where a ≠ 0, using the quadratic formula .
The calculator solution will show work using the quadratic formula to solve the entered equation for real and complex roots. Calculator determines whether the discriminant \( (b^2 - 4ac) \) is less than, greater than or equal to 0.
When \( b^2 - 4ac = 0 \) there is one real root.
When \( b^2 - 4ac > 0 \) there are two real roots.
When \( b^2 - 4ac < 0 \) there are two complex roots.
Quadratic Formula:
The quadratic formula
is used to solve quadratic equations where a ≠ 0 (polynomials with an order of 2)
Examples using the quadratic formula
Example 1: Find the Solution for \( x^2 + -8x + 5 = 0 \), where a = 1, b = -8 and c = 5, using the Quadratic Formula.
The discriminant \( b^2 - 4ac > 0 \) so, there are two real roots.
Simplify the Radical:
Simplify fractions and/or signs:
which becomes
Example 2: Find the Solution for \( 5x^2 + 20x + 32 = 0 \), where a = 5, b = 20 and c = 32, using the Quadratic Formula.
The discriminant \( b^2 - 4ac < 0 \) so, there are two complex roots.
calculator updated to include full solution for real and complex roots
Cite this content, page or calculator as:
Furey, Edward " Quadratic Formula Calculator " at https://www.calculatorsoup.com/calculators/algebra/quadratic-formula-calculator.php from CalculatorSoup, https://www.calculatorsoup.com - Online Calculators
Quadratic Equation Solver
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Example: 3x^2-2x-1=0
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Example: 3x^2-2x-1=0 (After you click the example, change the Method to 'Solve By Completing the Square'.)
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Example: 2x^2=18
Quadratic Formula
Example: 4x^2-2x-1=0
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- ax^2+bx+c=0
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Frequently Asked Questions (FAQ)
How do you calculate a quadratic equation.
- To solve a quadratic equation, use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
What is the quadratic formula?
- The quadratic formula gives solutions to the quadratic equation ax^2+bx+c=0 and is written in the form of x = (-b ± √(b^2 - 4ac)) / (2a)
Does any quadratic equation have two solutions?
- There can be 0, 1 or 2 solutions to a quadratic equation. If the discriminant is positive there are two solutions, if negative there is no solution, if equlas 0 there is 1 solution.
What is quadratic equation in math?
- In math, a quadratic equation is a second-order polynomial equation in a single variable. It is written in the form: ax^2 + bx + c = 0 where x is the variable, and a, b, and c are constants, a ≠ 0.
How do you know if a quadratic equation has two solutions?
- A quadratic equation has two solutions if the discriminant b^2 - 4ac is positive.
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Quadratic Equation Solver
We can help you solve an equation of the form " ax 2 + bx + c = 0 " Just enter the values of a, b and c below :
Is it Quadratic?
Only if it can be put in the form ax 2 + bx + c = 0 , and a is not zero .
The name comes from "quad" meaning square, as the variable is squared (in other words x 2 ).
These are all quadratic equations in disguise:
How Does this Work?
The solution(s) to a quadratic equation can be calculated using the Quadratic Formula :
The "±" means we need to do a plus AND a minus, so there are normally TWO solutions !
The blue part ( b 2 - 4ac ) is called the "discriminant", because it can "discriminate" between the possible types of answer:
- when it is positive, we get two real solutions,
- when it is zero we get just ONE solution,
- when it is negative we get complex solutions.
Learn more at Quadratic Equations
Note: you can still access the old version here .
Quadratic Formula Calculator
The calculator below solves the quadratic equation of ax 2 + bx + c = 0 .
In algebra, a quadratic equation is any polynomial equation of the second degree with the following form:
ax 2 + bx + c = 0
where x is an unknown, a is referred to as the quadratic coefficient, b the linear coefficient, and c the constant. The numerals a , b , and c are coefficients of the equation, and they represent known numbers. For example, a cannot be 0, or the equation would be linear rather than quadratic. A quadratic equation can be solved in multiple ways, including factoring, using the quadratic formula, completing the square, or graphing. Only the use of the quadratic formula, as well as the basics of completing the square, will be discussed here (since the derivation of the formula involves completing the square). Below is the quadratic formula, as well as its derivation.
Derivation of the Quadratic Formula
From this point, it is possible to complete the square using the relationship that:
x 2 + bx + c = (x - h) 2 + k
Continuing the derivation using this relationship:
Recall that the ± exists as a function of computing a square root, making both positive and negative roots solutions of the quadratic equation. The x values found through the quadratic formula are roots of the quadratic equation that represent the x values where any parabola crosses the x-axis. Furthermore, the quadratic formula also provides the axis of symmetry of the parabola. This is demonstrated by the graph provided below. Note that the quadratic formula actually has many real-world applications, such as calculating areas, projectile trajectories, and speed, among others.
Online Quadratic Formula Calculator
Apply the quadratic formula using wolfram|alpha, a useful tool for finding the solutions to quadratic equations.
Wolfram|Alpha can apply the quadratic formula to solve equations coercible into the form . In doing so, Wolfram|Alpha finds both the real and complex roots of these equations. It can also utilize other methods helpful to solving quadratic equations, such as completing the square, factoring and graphing.
Learn more about:
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Tips for entering queries
Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask about finding roots of quadratic equations.
- quadratic formula 4x^2 + 4 x - 8
- quadratic formula a = 1, b = -1, c = 2
- solve x^2 - x - 4 = 0
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What are quadratic equations, and what is the quadratic formula?
A quadratic is a polynomial of degree two..
Quadratic equations form parabolas when graphed, and have a wide variety of applications across many disciplines. In physics, for example, they are used to model the trajectory of masses falling with the acceleration due to gravity.
Situations arise frequently in algebra when it is necessary to find the values at which a quadratic is zero. In other words, it is necessary to find the zeros or roots of a quadratic, or the solutions to the quadratic equation. Relating to the example of physics, these zeros, or roots, are the points at which a thrown ball departs from and returns to ground level.
One common method of solving quadratic equations involves expanding the equation into the form and substituting the , and coefficients into a formula known as the quadratic formula. This formula, , determines the one or two solutions to any given quadratic. Sometimes, one or both solutions will be complex valued.
Discovered in ancient times, the quadratic formula has accumulated various derivations, proofs and intuitions explaining it over the years since its conception. Some involve geometric approaches. Others involve analysis of extrema. There are also many others. Those listed and more are often topics of study for students learning the process of solving quadratic equations and finding roots of equations in general.
Alternative methods for solving quadratic equations do exist. Completing the square, factoring and graphing are some of many, and they have use cases—but because the quadratic formula is a generally fast and dependable means of solving quadratic equations, it is frequently chosen over the other methods.
Quadratic Formula Calculator
Enter the equation you want to solve using the quadratic formula.
The Quadratic Formula Calculator finds solutions to quadratic equations with real coefficients. For equations with real solutions, you can use the graphing tool to visualize the solutions.
Quadratic Formula : x = − b ± b 2 − 4 a c 2 a
Click the blue arrow to submit. Choose "Solve Using the Quadratic Formula" from the topic selector and click to see the result in our Algebra Calculator !
Solve Using the Quadratic Formula Apply the Quadratic Formula
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Solve Using the Quadratic Formula x 2 + 5 x + 6 = 0 Solve Using the Quadratic Formula x 2 - 9 = 0 Solve Using the Quadratic Formula 5 x 2 - 7 x - 3 = 0 Apply the Quadratic Formula x 2 - 14 x + 49 Apply the Quadratic Formula x 2 - 18 x - 4
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Quadratic Equations
Solving equations is the central theme of algebra. All skills learned lead eventually to the ability to solve equations and simplify the solutions. In previous chapters we have solved equations of the first degree. You now have the necessary skills to solve equations of the second degree, which are known as quadratic equations .
QUADRATICS SOLVED BY FACTORING
- Identify a quadratic equation.
- Place a quadratic equation in standard form.
- Solve a quadratic equation by factoring.
A quadratic equation is a polynomial equation that contains the second degree, but no higher degree, of the variable.
The standard form of a quadratic equation is ax 2 + bx + c = 0 when a ≠ 0 and a, b, and c are real numbers.
All quadratic equations can be put in standard form, and any equation that can be put in standard form is a quadratic equation. In other words, the standard form represents all quadratic equations.
The solution to an equation is sometimes referred to as the root of the equation.
An important theorem, which cannot be proved at the level of this text, states "Every polynomial equation of degree n has exactly n roots." Using this fact tells us that quadratic equations will always have two solutions. It is possible that the two solutions are equal.
The simplest method of solving quadratics is by factoring. This method cannot always be used, because not all polynomials are factorable, but it is used whenever factoring is possible.
The method of solving by factoring is based on a simple theorem.
If AB = 0, then either A = 0 or B = 0.
We will not attempt to prove this theorem but note carefully what it states. We can never multiply two numbers and obtain an answer of zero unless at least one of the numbers is zero. Of course, both of the numbers can be zero since (0)(0) = 0.
Solution Step 1 Put the equation in standard form.
Step 2 Factor completely.
Step 3 Set each factor equal to zero and solve for x. Since we have (x - 6)(x + 1) = 0, we know that x - 6 = 0 or x + 1 = 0, in which case x = 6 or x = - 1.
Step 4 Check the solution in the original equation. If x = 6, then x 2 - 5x = 6 becomes
Therefore, x = 6 is a solution. If x = - 1, then x 2 - 5x = 6 becomes
Therefore, - 1 is a solution.
The solutions can be indicated either by writing x = 6 and x = - 1 or by using set notation and writing {6, - 1}, which we read "the solution set for x is 6 and - 1." In this text we will use set notation.
Check the solutions in the original equation.
INCOMPLETE QUADRATICS
- Identify an incomplete quadratic equation.
- Solve an incomplete quadratic equation.
If, when an equation is placed in standard form ax 2 + bx + c = 0, either b = 0 or c = 0, the equation is an incomplete quadratic .
5x 2 - 10 = 0 is an incomplete quadratic, since the middle term is missing and therefore b = 0.
When you encounter an incomplete quadratic with c - 0 (third term missing), it can still be solved by factoring.
Notice that if the c term is missing, you can always factor x from the other terms. This means that in all such equations, zero will be one of the solutions. An incomplete quadratic with the b term missing must be solved by another method, since factoring will be possible only in special cases.
Example 3 Solve for x if x 2 - 12 = 0.
Solution Since x 2 - 12 has no common factor and is not the difference of squares, it cannot be factored into rational factors. But, from previous observations, we have the following theorem.
Using this theorem, we have
Note that in this example we have the square of a number equal to a negative number. This can never be true in the real number system and, therefore, we have no real solution.
COMPLETING THE SQUARE
- Identify a perfect square trinomial.
- Complete the third term to make a perfect square trinomial.
- Solve a quadratic equation by completing the square.
From your experience in factoring you already realize that not all polynomials are factorable. Therefore, we need a method for solving quadratics that are not factorable. The method needed is called "completing the square."
First let us review the meaning of "perfect square trinomial." When we square a binomial we obtain a perfect square trinomial. The general form is (a + b) 2 = a 2 + 2ab + b 2 .
- The other term is either plus or minus two times the product of the square roots of the other two terms.
The -7 term immediately says this cannot be a perfect square trinomial. The task in completing the square is to find a number to replace the -7 such that there will be a perfect square.
Consider this problem: Fill in the blank so that "x 2 + 6x + _______" will be a perfect square trinomial. From the two conditions for a perfect square trinomial we know that the blank must contain a perfect square and that 6x must be twice the product of the square root of x 2 and the number in the blank. Since x is already present in 6x and is a square root of x 2 , then 6 must be twice the square root of the number we place in the blank. In other words, if we first take half of 6 and then square that result, we will obtain the necessary number for the blank.
Therefore x 2 + 6x + 9 is a perfect square trinomial.
Now let's consider how we can use completing the square to solve quadratic equations.
Example 5 Solve x 2 + 6x - 7 = 0 by completing the square.
Solution First we notice that the -7 term must be replaced if we are to have a perfect square trinomial, so we will rewrite the equation, leaving a blank for the needed number.
At this point, be careful not to violate any rules of algebra. For instance, note that the second form came from adding +7 to both sides of the equation. Never add something to one side without adding the same thing to the other side.
Now we find half of 6 = 3 and 3 2 = 9, to give us the number for the blank. Again, if we place a 9 in the blank we must also add 9 to the right side as well.
Now factor the perfect square trinomial, which gives
Example 6 Solve 2x 2 + 12x - 4 = 0 by completing the square.
Solution This problem brings in another difficulty. The first term, 2x 2 , is not a perfect square. We will correct this by dividing all terms of the equation by 2 and obtain
We now add 2 to both sides, giving
Example 7 Solve 3x 2 + 7x - 9 = 0 by completing the square.
Solution Step 1 Divide all terms by 3.
Step 2 Rewrite the equation, leaving a blank for the term necessary to complete the square.
Step 3 Find the square of half of the coefficient of x and add to both sides.
Step 4 Factor the completed square.
Step 5 Take the square root of each side of the equation.
Step 6 Solve for x (two values).
Follow the steps in the previous computation and then note especially the last ine. What is the conclusion when the square of a quantity is equal to a negative number? "No real solution."
In summary, to solve a quadratic equation by completing the square, follow this step-by-step method.
Step 1 If the coefficient of x2 is not 1, divide all terms by that coefficient. Step 2 Rewrite the equation in the form of x2 + bx + _______ = c + _______. Step 3 Find the square of one-half of the coefficient of the x term and add this quantity to both sides of the equation. Step 4 Factor the completed square and combine the numbers on the right-hand side of the equation. Step 5 Find the square root of each side of the equation. Step 6 Solve for x and simplify. If step 5 is not possible, then the equation has no real solution.
THE QUADRATIC FORMULA
- Solve the general quadratic equation by completing the square.
- Solve any quadratic equation by using the quadratic formula.
The standard form of a quadratic equation is ax 2 + bx + c = 0. This means that every quadratic equation can be put in this form. In a sense then ax 2 + bx + c = 0 represents all quadratics. If you can solve this equation, you will have the solution to all quadratic equations.
We will solve the general quadratic equation by the method of completing the square.
To use the quadratic formula you must identify a, b, and c. To do this the given equation must always be placed in standard form.
Not every quadratic equation will have a real solution.
There is no real solution since -47 has no real square root.
This solution should now be simplified.
WORD PROBLEMS
- Identify word problems that require a quadratic equation for their solution.
- Solve word problems involving quadratic equations.
Certain types of word problems can be solved by quadratic equations. The process of outlining and setting up the problem is the same as taught in chapter 5, but with problems solved by quadratics you must be very careful to check the solutions in the problem itself. The physical restrictions within the problem can eliminate one or both of the solutions.
Example 1 If the length of a rectangle is 1 unit more than twice the width, and the area is 55 square units, find the length and width.
Solution The formula for the area of a rectangle is Area = Length X Width. Let x = width, 2x + 1 = length.
At this point, you can see that the solution x = -11/2 is not valid since x represents a measurement of the width and negative numbers are not used for such measurements. Therefore, the solution is
width = x = 5, length = 2x + 1 = 11.
Example 3 If a certain integer is subtracted from 6 times its square, the result is 15. Find the integer.
Solution Let x = the integer. Then
Since neither solution is an integer, the problem has no solution.
Example 4 A farm manager has 200 meters of fence on hand and wishes to enclose a rectangular field so that it will contain 2,400 square meters in area. What should the dimensions of the field be?
Solution Here there are two formulas involved. P = 2l + 2w for the perimeter and A = lw for the area. First using P = 2l + 2w, we get
We can now use the formula A = lw and substitute (100 - l) for w, giving
The field must be 40 meters wide by 60 meters long.
Note that in this problem we actually use a system of equations
P = 2 l + 2 w A = l w.
In general, a system of equations in which a quadratic is involved will be solved by the substitution method. (See chapter 6.)
- A quadratic equation is a polynomial equation in one unknown that contains the second degree, but no higher degree, of the variable.
- The standard form of a quadratic equation is ax 2 + bx + c = 0, when a ≠ 0.
- An incomplete quadratic equation is of the form ax 2 + bx + c = 0, and either b = 0 or c = 0.
- The most direct and generally easiest method of finding the solutions to a quadratic equation is factoring. This method is based on the theorem: if AB = 0, then A = 0 or B = 0. To use this theorem we put the equation in standard form, factor, and set each factor equal to zero.
- To solve a quadratic equation by completing the square, follow these steps: Step 1 If the coefficient of x 2 is not 1, divide all terms by that coefficient. Step 2 Rewrite the equation in the form of x 2 + bx +_____ = c + _____ Step 3 Find the square of one-half of the coefficient of the x term and add this quantity to both sides of the equation. Step 4 Factor the completed square and combine the numbers on the right-hand side of the equation. Step 5 Find the square root of each side of the equation. Step 6 Solve for x and simplify.
- The method of completing the square is used to derive the quadratic formula.
- To use the quadratic formula write the equation in standard form, identify a, b, and c, and substitute these values into the formula. All solutions should be simplified.
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Example 1: Find the Solution for x2+−8x+5=0, where a = 1, b = -8 and c = 5, using the Quadratic Formula. ... The discriminant b2−4ac>0 so, there are two real
Shows you the step-by-step solutions using the quadratic formula! This calculator will solve your problems.
Step-By-Step Example. Learn step-by-step how to solve quadratic equations! · Example (Click to try) · Choose Your Method. There are different methods you can use
To solve a quadratic equation, use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a). What is the quadratic formula? The quadratic formula gives solutions
Quadratic Equation Solver. We can help you solve an equation of the form "ax2 + bx + c = 0" ... These are all quadratic equations in disguise:
A quadratic equation can be solved in multiple ways, including factoring, using the quadratic formula, completing the square, or graphing. Only the use of the
A useful tool for finding the solutions to quadratic equations ... Wolfram|Alpha can apply the quadratic formula to solve equations coercible into the form ax2+bx
Enter the equation you want to solve using the quadratic formula. The Quadratic Formula Calculator finds solutions to quadratic equations with real coefficients
The most direct and generally easiest method of finding the solutions to a quadratic equation is factoring. This method is based on the theorem: if AB = 0, then
Learn about quadratic equations using our free math solver with step-by-step solutions.