How to solve quadratic equations in the simplest way possible

Algebraic Process: Quadratic Equation

By the end of this post, you should be able to solve quadratic equations by:

i. Factorisation

ii. Completing the square

iii. Quadratic formula.

Consider this;

If the product of two real numbers is 0, then one of the number or both of them must be 0.

In general,

If a x b = 0

Either a = 0 or b = 0

Example 1

Solve the equation (x - 4)(x + 9) = 0

If (x - 4)(x + 9) = 0

Then either 

x - 4 = 0    Or     x + 9 = 0

x = 4         Or     x = -9

Example 2: Using Factorisation

Solve 2x² + 13x = 15

First thing here is to equate it zero by carrying +15 to LHS (Left Hand Side)

2x² + 13x - 15 = 0

If you notice, +15 changed to -15 when carried to LHS because, it's crossed the "=" sign.

Now, the above equation can be solved in two ways;

factorisation or quadratic equation

2x² + 13x - 15 = 0

Can be factorised to get

2x² + 15x - 2x - 15 = 0

x(2x + 15) -1(2x + 15) = 0

(2x + 15)(x - 1) = 0

Then,

2x + 15 = 0          or      x - 1 = 0

2x = -15               or      x = 1

x = (-15/2)

x = -7½                or      x = 1

Quadratic Equation

Before we delve into quadratic equation, we must shed a little light on completing the square

Completing the square

Example 

What must be added to d² - 5d to make the expression a perfect square?

Solution

The coefficient of d is -5. 

Divide -5 by 2 which is -5/2.

(-5/2)²

= +25/4

That means, +25/4 must be added to d² - 5d to make it a perfect square.

d² - 5d + 25/4

It becomes simpler when you go over it again. Probably these steps might help;

i. Find the coefficient of the non-squared term

ii. Divide it by 2

iii. Add the result to the question....Viola

Back to quadratic equation

The general form of a quadratic equation is

ax² + bx + c = 0. The  roots of the general quadratic equation are found by completing the square.

Check the image below:

This is called the quadratic formula. Use the formula to solve any quadratic equation.

Consider this example

Find the roots of the equation 3x² - 5x - 7 = 0.

Check the image below for simplified calculation

That's all for now, please comment and share your thoughts. Any question will surely be answered.

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