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Unit 4:MathematicsAims• Introduce

Quadratic Formulae.

Objectives• Identify how to solve

quadratic equation’s using a quadratic equation.

Multiplying out brackets

Brackets can be removed from an expression by multiplying out brackets

e.g. (x + 5)(x + 7) = x2 + 7x + 5x + 35 = x2 + 12x + 35

This give a quadratic expression. A quadratic expression looks like ax2 + bx + c where a, b and c are numbers and a must not equal 0.

x +5

x

+7

x2 + 5x

+ 7x + 35

2 4

2

b b acx

a

The Quadratic Equation!

Used to solve an equation in the form:ax2 + bx + c = 0

2 4

2

b b acx

a

x2 + x - 2 = 0

a= 1, b = 1, c = -2

21 1 4 1 2

2 1x

1 1 8

2x

1 1 8

2x

1 9

2x

1 3

2x

1 3 21

2 2x

1 3 42

2 2x

x = 1 or -2

2 4

2

b b acx

a

2x2 + 5x - 3 = 0

a= 2, b = 5, c = -3

x = -3 or 0.5

25 5 4 2 3

2 2

5 25 24

4

5 25 24

4

5 49

4

5 7

4

5 7 20.5

4 4

5 7 123

4 4

Solve the quadratic equation 3x² + 9x+4 = 0

Here a = 3, b = 9 and c = 4. Putting these values into the quadratic formula gives

Solve the quadratic equation 8x² + 3x − 4 = 0 .Care is needed here because the value of c is negative, that is c = −4.

Find the roots of the following quadratic equations:a)x² + 6x − 8 = 0 , b)2x² − 8x − 3 = 0 , c)-3x² + x+1 = 0 .

• A cyclist takes (x – 21) hours to travel a distance of 46 kilometres when cycling at an average speed of x kilometres. Find the cyclist average speed?

Distance = Speed x Time

46 = x(x-21)46 =x² - 21