1. Quadratics
Friday, 26 April 2019

2. Solving Quadratics Example Solve each of the following quadratics
b) 𝑡 2 −6𝑡=40
c) 10 𝑥 2 =𝑥+2
b) 𝑡 2 −6𝑡=40
c) 10 𝑥 2 =𝑥+2
a) 𝑥 2 −11𝑥+24=0
𝑡 2 −6𝑡−40=0
10 𝑥 2 −𝑥−2=0
𝑥−3
𝑥−8 =0
𝑡−10
𝑡+4 =0
5𝑥+2
2𝑥−1 =0
𝑥=3
𝑥=8
𝑡=10
𝑡=−4
𝑥=− 2 5
𝑥= 1 2

3. Example
Show that the area of the shape drawn below can be given as 𝑥 2 +8𝑥+15
b) Hence given that the area is equal to 48 cm2 find the possible value of x.

4. A B a) Area A =𝑥 𝑥+2 =𝑥 2 +2𝑥 Area B =3 2𝑥+5 =6𝑥+15 Area = 𝑥 2 +8𝑥+15
𝑥 2 +8𝑥+15=48
𝑥 2 +8𝑥−33=0
𝑥−3
𝑥+11 =0
𝑥=3
𝑥=−11
∴𝑥=3

5. Algebraic Fractions 𝑥 2 −3𝑥 𝑥 2 −2𝑥−3 𝑥 2 −3𝑥 𝑥 2 −2𝑥−3
Example
Factorise 𝑥 2 −3𝑥
Factorise 𝑥 2 −2𝑥−3
Hence simplify the fraction
𝑥 2 −3𝑥 𝑥 2 −2𝑥−3
a) 𝑥 2 −3𝑥
=𝑥 𝑥−3
b) 𝑥 2 −2𝑥−3
= 𝑥−3
𝑥+1
𝑥 2 −3𝑥 𝑥 2 −2𝑥−3
= 𝑥 𝑥−3 𝑥−3 𝑥+1
= 𝑥 𝑥+1 c)

6. 𝑥 2 +4𝑥 2 𝑥 2 −10𝑥 = 𝑥 𝑥+4 2𝑥 𝑥−5 = 𝑥+4 2 𝑥−5 𝑥 2 +6𝑥+5 𝑥 2 −𝑥−2
Example
Write each of the following fractions in their simplest form
𝑥 2 +4𝑥 2 𝑥 2 −10𝑥
= 𝑥 𝑥+4 2𝑥 𝑥−5
= 𝑥+4 2 𝑥−5
(i)
𝑥 2 +6𝑥+5 𝑥 2 −𝑥−2
= 𝑥+1 𝑥+5 𝑥+1 𝑥−2
= 𝑥+5 𝑥−2
(ii)

7. In order to simplify an Algebraic fraction
Ensure that both numerator and denominator are factorised
Cancel any factors that are common to both numerator and denominator
Example
Write the following fractions in its simplest form
4 𝑥 2 +8𝑥 8 𝑥 2 −24𝑥
= 4𝑥 𝑥+2 8𝑥 𝑥−3
= 𝑥+2 2 𝑥−3 2

8. 1. Write each of the following fractions in their simplest form
Questions
1. Write each of the following fractions in their simplest form
Answer =
(i)
(ii)
Answer =
(iii)
Answer =

9. (iv)
Answer =
(v)
Answer =
(vi)
Answer =

10. 2. solve each of the following equations
b) 3 𝑥 2 +4𝑥+1=0
c) 4 𝑥 2 −4𝑥−15=0
d) 10 𝑥 2 −𝑥−2=0
e) 4 𝑥 2 =6−5𝑥
Answer
Answer
Answer
Answer
Answer

11. 3. The base of a triangle is 7 cm longer than its height
3. The base of a triangle is 7 cm longer than its height. The area of the triangle is 30 cm2.
(a) Taking the height to be ℎ cm, show that
ℎ2+ 7ℎ − 60 = 0
(b) Solve this equation to find the height of the triangle.
Answer cm