How to Solve Quadratic Equations?

There are basically four methods of solving quadratic equations. They are:

1. Factoring
2. Completing the square
3. Using Quadratic Formula
4. Taking the square root

Factoring of Quadratics

• Begin with a equation of the form ax² + bx + c = 0
• Ensure that it is set to adequate zero.
• Factor the left-hand side of the equation by assuming zero on the right-hand side of the equation.
• Assign each factor equal to zero.
• Now solve the equation in order to determine the values of x.

Suppose if the main coefficient is not equal to one then deliberately, you have to follow a methodology in the arrangement of the factors.

Example:

2x²-x-6=0

(2x+3)(x-2)=0

2x+3=0

x=-3/2

x=2

Completing the Square Method

Let us learn this method with example.

Example: Solve 2x² – x – 1 = 0.

First, move the constant term to the other side of the equation.

2x² – x = 1

Dividing both sides by 2.

x² – x/2 = ½

Add the square of half of the coefficient of x, (b/2a)2, on both the sides, i.e., 1/16

x² – x/2 + 1/16 = ½ + 1/16

Now we can factor the right side,

(x-¼)2 = 9/16 = (¾)2

Taking root on both sides;

X – ¼ = ±3/4

Add ¼ on both sides

X = ¼ ± ¾

Therefore,

X = ¼ + ¾ = 4/4 = 1

X = ¼ – ¾ = -2/4 = -½ 

Using Quadratic Formula

For the given Quadratic equation of the form, ax² + bx + c = 0

Therefore the roots of the given equation can be found by:

where ± (one plus and one minus) represent two distinct roots of the given equation.

Taking the Square Root

We can use this method for the equations such as:

x² + a² = 0

Example: Solve x² – 50 = 0.

x² – 50 = 0

x² = 50

Taking the roots both sides

√x² = ±√50

x = ±√(2 x 5 x 5)

x = ±5√2

Thus, we got the required solution.