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Calculators :: Solving Equations :: Quadratic Equation Solver

This calculator solves quadratic equations. You can solve equations by completing the square or by using quadratic formula. The calculator will write down complete solution. Every step will be explained in detail.

Quadratic Equation Solver

Enter quadratic equation of the form ( ax2 + bx + c = 0 )
You can enter either integers (10), decimal numbers(10.12) or FRACTIONS (10/3).
Important: The form will NOT let you enter illegal characters like a, u, %, y, p, ... How to input?? 

Solve by completing the square (default)
Solve by using quadratic formula

How to solve quadratic equation ?

A general quadratic equation can be written in the form $ax^2 + bx + c = 0$. This calculator solves quadratic equation using two methods.

Method 1: Use the Quadratic Formula

When $a \ne 0$ , there are two solutions to $ax^2 + bx + c = 0$ and they are

$$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}.$$

The formula can be used to solve any quadratic equation.

Example:

Solve equation $2x^2 + 7x - 15 = 0$ using the quadratic formula.

Solution:

Here we have : $ a = 2 ~ b = 7 ~ c = -15 $

$$x_{1,2} = \frac{ -b \pm \sqrt{b^2-4 a c} }{2a}$$
$$x_{1,2} = \frac{ -7 \pm \sqrt{7^2 - 4\cdot 2\cdot(-15)}}{2\cdot2}$$
$$x_{1,2} = \frac{ -7 \pm \sqrt{49 + 120}} {4}$$
$$x_{1,2} = \frac{ -7 \pm \sqrt{169}} {4}$$
$$x_{1,2} = \frac{ -7 \pm 13} {4}$$

To calculate first solution we use "+" sign:

To calculate second solution we use "-" sign:

$$x_{1} = \frac{-7 + 13} {4}$$
$$x_{1} = \frac{6}{4}$$
$$x_{1} = \frac{3}{2}$$
$$x_{2} = \frac{-7 - 13} {4}$$
$$x_{2} = \frac{-20} {4}$$
$$x_{2} = -5$$

Exercise:

Solve equation 3x2 + 2x - 5 = 0. ( Use above calculator to check your solution. )

 

Method 2: Completing the square

The best way to learn this method is by using an example.

Example:

Solve equation 2x2 + 7x - 15 = 0 by completing the square.

Solution:

$$2x^2 + 7x - 15 = 0$$
$$2x^2 + 7x - 15 = 0 / : 2$$
$$x^2 + \frac{7}{2}x - \frac{15}{2} = 0$$

Step1: Divide all terms by the coefficient of x2.

$$x^2 + \frac{7}{2}x = \frac{15}{2}$$

Step 2: Keep all terms containing x on one side. Move the constant to the right.

$$x^2 + \frac{7}{2}x + {\left(\frac{7}{4}\right)}^2= \frac{15}{2}+{\left(\frac{7}{4}\right)}^2$$

Step 3: Take half of the x-term coefficient and square it. Add this value to both sides.

$$x^2 + \frac{7}{2}x + {\left(\frac{7}{4}\right)}^2= \frac{169}{16}$$

Step 4: Simplify right side.

$$ {\left(x + \frac{7}{4}\right)}^2= \frac{169}{16}$$

Step 5: Write the perfect square on the left.

$$ x + \frac{7}{4}= \pm\sqrt{\frac{169}{16}}$$
$$ x + \frac{7}{4}= \pm\frac{13}{4}$$

Step 6: Take the square root on both sides of the equation.

$$ x_1 = - \frac{7}{4} + \frac{13}{4} = \frac{3}{2}$$
$$ x_2 = - \frac{7}{4} - \frac{13}{4} = -5$$

Step 7: Solve for x.

Exercise:

Solve equation x2 - 4x + 3 = 0. ( Use above calculator to check your solution. )

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