Quadratic Equations

Prepared by Golam Robbani Ahmed

An example of a Quadratic Equation:

The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x²).

It is also called an "Equation of Degree 2" (because of the "2" on the x)

The Standard Form of a Quadratic Equation looks like this:

Where a, b and c are known values. a can't be 0."x" is the variable or unknown

What is solution of Quadratic equation?

The solution of Quadratic Equation means finding the variable. There are usually 2 solutions . They are also called "roots", or sometimes "zeros“.There are Four methods of solutions1.Factorization method2.By completing Square3.By Quadratic formula4.Graphing Method

Factorisation Method•Factoring is typically one of the easiest and quickest ways to solve quadratic equations;

•However,all quadratic polynomials can not be factorised.

•This means that factoring will not work to solve many quadratic equations.

Factoring (Examples)•Example 1•x2 – 2x – 24 = 0

•(x + 4)(x – 6) = 0

•x + 4 = 0 x – 6 = 0

• x = –4 x = 6

Example 2x2 – 8x + 11 = 0

x2 – 8x + 11 is prime; therefore, another method must be used to solve this equation.

Solve by factorization

X=1/4

Solve by factorization= 0

X=-5/√2 or x=--√2

Derivation of Quadratic FormulaA Quadratic Equation looks like this:

That formula looks like magic, but you can follow the steps to see how it comes about.

aacbbxcbxax

240

22

If we have a quadratic equation and are considering solutions from the real number system, using the quadratic formula, one of three things can happen.

3. The "stuff" under the square root can be negative and we'd get no real solutions.The "stuff" under the square root is called the discriminant.

This "discriminates" or tells us what type of solutions we'll have.

1. The "stuff" under the square root can be positive and we'd get two unequal real solutions 04 if 2 acb2. The "stuff" under the square root can be zero and we'd get one solution (called a repeated or double root because it would factor into two equal factors, each giving us the same solution).04 if 2 acb

04 if 2 acb

The Discriminant acb 42

Square Root Property•This method is also relatively quick and easy;

•It only works for equations in which the quadratic polynomial is written in the following form.

• x2 = n or (x + c)2 = n

Square Root Property (Examples)Example 1 Example 2

x2 = 49 (x + 3)2 = 25

x = ± 7 x + 3 = ± 5

x + 3 = 5 x + 3 = –5

x = 2 x = –8

2 49x 2( 3) 25x

By completing the square

Completing the Square (Examples

•Example 1

•a = 1, b is even

•x2 – 6x + 13 = 0•x2 – 6x + 9 = –13 + 9• (x – 3)2 = –4 • x – 3 = ± 2i

• x = 3 ± 2i

Example 2

a ≠ 1, b is not even

3x2 – 5x + 2 = 0

2 5 2 03 3x x

2 5 25 2 253 36 3 36x x

25 16 36x

5 16 6x

5 16 6x

5 16 6x

OR

x = 1 OR x = ⅔

Quadratic Formula•This method will work to solve all quadratic equations;

•However,for many equations it takes longer than some of the methods discussed earlier.

•The quadratic formula is a good choice if the quadratic polynomial cannot be factorised

Quadratic Formula (Example)•x2 – 8x – 17 = 0•In this method •equation is comparedWith standard quadraticequation

•a = 1•b = –8•c = –17

28 ( 8) 4(1)( 17)2(1)

x

8 64 682

x

8 1322

x

8 2 332

x

4 33

Graphing•Graphing to solve quadratic equations does not always produce an accurate result.

•If the solutions to the quadratic equation are irrational or complex, there is no way to tell what the exact solutions are by looking at a graph.

•Graphing is very useful when solving contextual problems involving quadratic equations.