SOLVING QUADRATICSSOLVING QUADRATICS

SOLVING QUADRATICSSOLVING QUADRATICS

•General Form:

cbxaxy 2

•Where a, b and c are constants

02 cbxax

To solve a quadratic equation, the equation must be expressed in the form:

That is, all variables and constants must be on one side of the equals sign with zero on the other.

02 cbxax

Methods for Solving Quadratic equations

•Method 2 : Quadratic Formula

a

acbbx

2

42

•Method 1 : Factorisation

•Method 4 : Graphic Calculator

•Method 3 : Completing the Square

•Method 1 : Factorisation

Example 1 : 020122 xx

020122 xx

0210 xx

0)2(0)10( xx or either

10x 2xor

Step 1: Equation in the right form

Step 2: Factorise

Step 3: Separate the two parts of the product

Step 4: Solve each equation

Example 1 : 020122 xx

020122 xx

0210 xx

02010 xx or either

10x 2xor

Some points to note

Need to remember your factorisation skills

The equation is expressed as the product of two factors being equal to zero, therefore, one (or both) of the factors must be zero.

To check your answers are correct you can substitute them one at a time into the original equation.

Example 1 : 020122 xx

10x

2x

Checking solutions

0

20120100

20)10(12102

0

20244

20)2(1222

•Method 1 : Factorisation

Example 2 : 2832 xx

02832 xx

047 xx

0407 xx or either

7x 4xor

Step 1: Equation in the right form

Step 2: Factorise

Step 3: Separate the two parts of the product

Step 4: Solve each equation

•Method 2 : Quadratic Formula

Example 1 : 0582 xxa

acbbx

2

42

)1(2

)5)(1(4)8(8 2 x

2

20648 x

2

848x

2

2128x

a = 1 b = – 8 and c = – 5

Step 1: Determine the values of a, b and c

Step 2: Substitute the values of a, b and c

Step 3: Simplify

See next slideThis step is optional. Students need to have covered surds

2

2128x

2

2142 x

214x

58.0214

58.8214

x

Exact answer

Decimal approximation

Note that 2 is a factor of the numerator

Once 2 is factored out, it can be cancelled with the 2 in the denominator

•Method 2 : Quadratic Formula

Example 2 : 0283 2 xxa

acbbx

2

42

)3(2

)2)(3(488 2 x

6

24648 x

6

408x

6

1028x

a = 3 b = 8 and c = 2

Step 1: Determine the values of a, b and c

Step 2: Substitute the values of a, b and c

Step 3: Simplify

See next slideThis step is optional. Students need to have covered surds

6

1042 x

3

104x

39.23

104

28.03

104

x

Exact answer

Decimal approximation

Note that 2 is a factor of the numerator

Once 2 is factored out, it can be cancelled with the 6 in the denominator

6

1028x

•Method 2 : Quadratic Formula

Example 3 : 0675 2 xxa

acbbx

2

42

)5(2

)6)(5(4)7(7 2 x

10

120497 x

10

717 x

a = 5 b = – 7 and c = 6

Step 1: Determine the values of a, b and c

Step 2: Substitute the values of a, b and c

Step 3: Simplify

Problem - you cannot find the square root of a negative number

2 Imaginary Solutions

That was a lot of work to find that there was no solution!

It would be useful to be able to “test” the equation before we start.

For this we use the DISCRIMINANT.

The Discriminant

The discriminant is a quick way to check how many real solutions exist for a given quadratic equation.

acb 42

As shown above the symbol for the discriminant is and it is calculated using .acb 42

Summary of Results using Discriminant

> 0 The equation has two real solutions

acb 42

= 0 The equation has one real solutions

< 0 The equation has no real solutions

Relating the Discriminant to graphs

> 0acb 42

The graph cuts the x-axis in two places. These are the 2 real solutions to the quadratic equation.

y)

x

y

x

y

x

Relating the Discriminant to graphs

acb 42 = 0

These graphs have their turning point on the x-axis and hence there are 2 equal solutions.

y)

x

y

x

y

x

Relating the Discriminant to graphs

acb 42 < 0

There are 2 imaginary solutions in this case because the graphs do not intersect with the x-axis.

y)

x

y

x

y

x

•Method 3: Completing the Square Technique

0362 xx

039)3( 2 x

06)3( 2 x

Example 1 :

This value is half b

Subtract the square of the number in the bracket

6)3( 2 x63 x

63 x

Add 6 to both sides

Take the square root of both sides

Subtract 3 from both sides

•Method 3: Completing the Square Technique

This result gives us the exact answers.

63 x

or63 x 63 xUse your calculator to find decimal approximations accurate to two decimal places.

45.555.0 xx or

•Method 3: Completing the Square Technique

0852 xx

084

25)2

5( 2 x

04

57)2

5( 2 x

Example 2 :

4

57)2

5( 2 x

2

575.2 x

2

575 x

Take the square root of both sides

Subtract 2.5 from both sides

This value is half b

Subtract the square of the number in the bracket

Add to both sides4

57

Example 3: Solve 01072 xx

01072 xx

0]25.2)5.3[( 2 x

01025.12)5.3( x

Factor out the coefficient of x²

025.2)5.3( 2 x

This value is half b

Subtract the square of the number in the bracket

25.2)5.3( 2 x25.2)5.3( 2 x

See next slide

25.2)5.3( 2 x

25.25.3 x

25.25.3 x

or i.e. 25.25.3 x 25.25.3 x

•Method 4 Graphic Calculator

Graph the function and find the value of the x-intercepts

Use a solver function for a polynomial of degree 2

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