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## Quadratic Formula Calculator

## Calculator Use

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:

is used to solve quadratic equations where a ≠ 0 (polynomials with an order of 2)

## Examples using the quadratic formula

The discriminant \( b^2 - 4ac > 0 \) so, there are two real roots.

Simplify fractions and/or signs:

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:

## Quadratic Equation Solver

What do you want to calculate.

## Step-By-Step Example

Example (click to try), choose your method, solve by factoring.

## Complete The Square

## Take the Square Root

## Quadratic Formula

## Quadratic Equation Calculator

Solve quadratic equations step-by-step.

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## Most Used Actions

## Frequently Asked Questions (FAQ)

How do you calculate a quadratic equation.

## What is the quadratic formula?

## Does any quadratic equation have two solutions?

## What is quadratic equation in math?

## How do you know if a quadratic equation has two solutions?

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## Quadratic Equation Solver

## 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 !

- 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 .

## Derivation of the Quadratic Formula

From this point, it is possible to complete the square using the relationship that:

Continuing the derivation using this relationship:

## Online Quadratic Formula Calculator

## Tips for entering queries

- quadratic formula 4x^2 + 4 x - 8
- quadratic formula a = 1, b = -1, c = 2
- solve x^2 - x - 4 = 0
- solve x^2 - 3x - 4 = 0
- View more examples »

## Access instant learning tools

Get immediate feedback and guidance with step-by-step solutions and Wolfram Problem Generator

## What are quadratic equations, and what is the quadratic formula?

A quadratic is a polynomial of degree two..

## Quadratic Formula Calculator

Enter the equation you want to solve using the quadratic formula.

Quadratic Formula : x = − b ± b 2 − 4 a c 2 a

Solve Using the Quadratic Formula Apply the Quadratic Formula

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## Quadratic Equations

## QUADRATICS SOLVED BY FACTORING

- Identify a quadratic equation.
- Place a quadratic equation in standard form.
- Solve a quadratic equation by factoring.

The solution to an equation is sometimes referred to as the root of the equation.

The method of solving by factoring is based on a simple theorem.

If AB = 0, then either A = 0 or B = 0.

Solution Step 1 Put the equation in standard form.

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

Check the solutions in the original equation.

## INCOMPLETE QUADRATICS

5x 2 - 10 = 0 is an incomplete quadratic, since the middle term is missing and therefore b = 0.

Example 3 Solve for x if x 2 - 12 = 0.

## 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.

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.

Now factor the perfect square trinomial, which gives

Example 6 Solve 2x 2 + 12x - 4 = 0 by completing the square.

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).

In summary, to solve a quadratic equation by completing the square, follow this step-by-step method.

## THE QUADRATIC FORMULA

- Solve the general quadratic equation by completing the square.
- Solve any quadratic equation by using the quadratic formula.

We will solve the general quadratic equation by the method of completing the square.

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.

width = x = 5, length = 2x + 1 = 11.

Solution Let x = the integer. Then

Since neither solution is an integer, the problem has no solution.

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

- 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.

## IMAGES

## VIDEO

## COMMENTS

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.