Quadratic equation solver

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 $

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. )