Quadratic Equations

Quadratic equations have one variable and the highest exponent is 2.

Quadratic equations can have zero, one or two solutions.

By factorising
By Taking the Square Root
By Graphing

quadratic_formula.jpgAt this level, there are three methods for solving quadratic equations.

By factorising.

Rearrange the equation so that all of the terms are on one side of the equation. Then factorise.

The method depends on the fact that if two factors multiplied together are equal to zero, then either one or both of them must be equal to zero.

 

Examples
Answers

Solve:

  

(a) x2 − 4x − 5 = 0

Factorise to (x − 5)(x + 1) = 0

Either: x − 5 = 0 or x + 1 = 0

x = 5 or x = -1

(b) (2x − 3)(x + 4) = 0

Already factorised

Either: 2x − 3 = 0 or x + 4 = 0

2x = 3 or x = − 4

x = 1.5

(c) 3x2 − 6x = 0

 

Factorise to 3x(x − 2) = 0

Either: 3x = 0 or x − 2 = 0

x = 0 or x = 2

(d) (x − 4)2 = 0

 

Already factorised (x − 4)(x − 4) = 0

Either: x − 4 = 0 or x − 4 = 0

x = 4

 



By taking the square root.

Take the square root of both sides of the equation.

Remember, if x2 = a then Y10_Quadratic_Equations_01.gif

 

Examples
Answers

(a) Solve x2 = 36

 

Take the square root of both sides.

x = ± 6

(b) Solve (x − 4)2 = 9

Take the square root of both sides.

x − 4 = ± 3

x = 7 or 1

 

By graphing.

The solutions for the equation are the x-intercepts of the parabola. See Topic 38