Quadratics Friday, 26 April 2019

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

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.

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

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)

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)

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

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 =

(iv) Answer = (v) Answer = (vi) Answer =

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

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