GCSE: Algebraic Fractions Skipton Girls’ High School

GCSE Specification 53ii. Simplify algebraic fractions. (Using revision pack reference numbers) 53ii. Simplify algebraic fractions. 53iii. Add and subtract algebraic fractions. 92. Solve equations involving algebraic fractions which lead to quadratic equations.

40 - x 3 40 3 = x + 4 = 2x + 4 2(4) = 5x - 2 2(4 – 2x) = 3x - 2 Starter (Click your answer) Are these algebraic steps correct? 40 - x 3 40 3 = x + 4 = 2x + 4 Fail  Win!  2(4) = 5x - 2 2(4 – 2x) = 3x - 2 Ask students why they think the step might have been made (in addition to pointing out why it’s incorrect!). Emphasise that when a term is contained within either a bracket, fraction or root, applying the reverse of the term to the whole expression doesn’t generally have the desired effect. Fail   Win! 2 =3𝑥+2 2−𝑥 =2𝑥+3  Fail Win! 

𝑎 2 𝑏 𝑎+𝑏 𝑎𝑏 𝑏 Starter   Are these algebraic steps correct? Fail (Click your answer) Are these algebraic steps correct? 𝑎 2 𝑏 𝑎+𝑏 𝑎𝑏 𝑏 Fail  Win! 

y2 + x s(4 + z) 2 + x s 𝑥 2 +2 =𝑦+2 1 + r (2x+1)(x – 2) pq(r+2) + 1 Starter To cancel or not to cancel, that is the question? (Click your answer) y2 + x 2 + x s(4 + z) s 𝑥 2 +2 =𝑦+2  Fail Win!  Fail  Win!  Fail  Win!  (2x+1)(x – 2) x – 2 pq(r+2) + 1 pq 1 + r 2 - 1 Fail  Win!  Fail  Win!  Fail  Win! 

What did we learn? 𝑎+𝑏 𝑐+𝑏 → 𝑎 𝑐  Tip #1: You can’t add or subtract a term which is ‘trapped’ inside a bracket, fraction or root. 2 𝑎−𝑥 =2𝑥+1 → 2 𝑎 =3𝑥+1  Tip #2: In a fraction, we can only divide top and bottom by something, not add/subtract. (e.g. 5 7 is not the same as 6 8 !) 𝑎+𝑏 𝑐+𝑏 → 𝑎 𝑐 

Simplifying Algebraic Fractions Tip: Just factorise top and bottom, then cancel! 2 𝑥 2 +4𝑥 𝑥 2 −4 = 2𝑥 𝑥−2 ? 3𝑥+3 𝑥 2 +3𝑥+2 = 3 𝑥+2 ? 2 𝑥 2 −5𝑥−3 6 𝑥 3 −2 𝑥 4 =− 2𝑥+1 2 𝑥 3 ?

𝑥(𝑥+1) 𝑥 2 −1 → 𝑥 𝑥−1 2 𝑥 2 +5𝑥−3 𝑥 2 −9 → 2𝑥−1 𝑥−3 Test Your Understanding Tip: Sometimes they’ve done part of the factorising for you! 𝑥(𝑥+1) 𝑥 2 −1 → 𝑥 𝑥−1 2 𝑥 2 +5𝑥−3 𝑥 2 −9 → 2𝑥−1 𝑥−3 𝑥 2 +2𝑥𝑦+ 𝑦 2 2𝑥+2𝑦 → 𝑥+𝑦 2 ? ? ?

− 4−𝑦 =𝑦−4 − 2𝑥−9 =9−2𝑥 1−𝑥 𝑥−1 =−1 3−2𝑥 2−𝑥 2𝑥−3 𝑥+1 = 𝑥−2 𝑥+1 Negating a difference − 4−𝑦 =𝑦−4 − 2𝑥−9 =9−2𝑥 1−𝑥 𝑥−1 =−1 3−2𝑥 2−𝑥 2𝑥−3 𝑥+1 = 𝑥−2 𝑥+1 ? ? ? ?

Exercise 1 𝑥+10 5 𝑥+10 4 =𝑥+10 ? 2𝑥+6 2𝑥 = 𝑥+3 𝑥 4𝑥+8 3𝑥+6 = 4 3 𝑥 2 +𝑥−6 𝑥 2 −7𝑥+10 = 𝑥+3 𝑥−5 2𝑥+10 𝑥 2 −25 = 2 𝑥−5 𝑝 2 −9 2𝑝+6 = 𝑝−3 2 𝑥 2 +𝑥−2 𝑥 2 −4 = 𝑥−1 𝑥−2 6 𝑥 2 +3𝑥 4 𝑥 2 −1 = 3𝑥 2𝑥−1 ? 1 8 ? 𝑥 2 +2𝑥+1 𝑥 2 +3𝑥+2 = 𝑥+1 𝑥+2 𝑥 2 −8𝑥+15 2 𝑥 2 −7𝑥−15 = 𝑥−3 2𝑥+3 𝑥 2 −9 2 𝑥 2 −7𝑥+3 = 𝑥+3 2𝑥−1 6 𝑥 2 −𝑥−1 4 𝑥 2 −1 = 3𝑥+1 2𝑥+1 2 𝑦 2 +4𝑦 3 𝑦 2 +7𝑦+2 × 9 𝑦 2 −1 3 𝑦 2 −𝑦 =2 ? 9 2 ? 3 10 ? ? 4 11 ? ? 5 12 ? ? 6 ? 7 ? 13

Adding/Subtracting Fractions What’s our usual approach for adding fractions? ? Sometimes we don’t need to multiply the denominators. We can find the Lowest Common Multiple of the denominators. ? ?

! Adding/Subtracting Algebraic Fractions ? ? The same principle can be applied to algebraic fractions. ! ? ? Bro Tip: Notice that with this one, we didn’t need to times x and x2 together: x2 is a multiple of both denominators.

Further Example 3 𝑥−1 − 4 𝑥 = 3𝑥−4 𝑥−1 𝑥 𝑥−1 = 3𝑥−4𝑥+4 𝑥 𝑥−1 = −𝑥+4 𝑥 𝑥−1 ? ? ? Bro Tip: Be careful with your negatives! Bro Tip: The numerator needn’t be expanded out because it is factorised – you get the marks either way.

Test Your Understanding ? 2 𝑥+1 + 3 𝑥−1 = 5𝑥+1 𝑥+1 𝑥−1 ? ? ? ? 2 𝑥−2 − 𝑥 𝑥+1 = 2− 𝑥 2 +4𝑥 𝑥−2 𝑥+1 “To learn the secret ways of the ninja, add fractions you must.”

Exercise 2 Write as a single fraction in its simplest form. 1 𝑥 + 1 𝑦 = 𝑦+𝑥 𝑥𝑦 1 𝑥+4 + 2 𝑥−4 = 3𝑥+4 𝑥+4 𝑥−4 𝑥+3 4 + 𝑥−5 3 = 7𝑥−11 12 2 𝑥+4 − 1 𝑥−4 = 𝑥−12 𝑥+4 𝑥−4 2 𝑥−1 − 1 𝑥+1 = 𝑥+3 𝑥−1 𝑥+1 4 2𝑥−1 − 3 2𝑥+1 = 2𝑥+7 2𝑥−1 2𝑥+1 2 𝑥+1 − 𝑥 𝑦 = 2𝑦− 𝑥 2 −1 𝑦 𝑥+1 ? 2 𝑦+3 − 1 𝑦−6 = 𝑦−15 𝑦+3 𝑦−6 2 𝑥 2 −9 + 1 𝑥+3 = 𝑥−1 𝑥+3 𝑥−3 2 2−𝑥 − 4 4−𝑥 = 2𝑥 2−𝑥 4−𝑥 3 𝑥+1 + 4 𝑥+1 2 = 3𝑥+7 𝑥+1 2 1 𝑥−3 − 2 3𝑥−1 = 𝑥+5 𝑥−3 3𝑥−1 1 𝑥 + 1 𝑥+1 + 1 𝑥+2 = 3 𝑥 2 +6𝑥+2 𝑥(𝑥+1)(𝑥+2) 1 𝑥+1 10 + 1 𝑥+1 9 = 𝑥+2 𝑥+1 10 ? 1 8 ? ? 9 2 ? ? 3 10 ? ? 11 4 ? ? 5 12 ? N1 ? 6 ? ? 7 N2

Solving Equations with Algebraic Fractions When asked to solve an equation with fractions: Combine fractions into single fraction then multiply through by denominator. But if multiplying everything by 𝑥 2 turns equation into a quadratic, this is simpler. 8 𝑥 2 + 4 𝑥 =4 8+4𝑥=4 𝑥 2 2+𝑥= 𝑥 2 𝑥 2 −𝑥−2=0 𝑥+1 𝑥−2 =0 𝒙=−𝟏 𝒐𝒓 𝒙=𝟐 𝑥 2𝑥−3 + 4 𝑥+1 =1 𝑥 2 +9𝑥−12 2𝑥−3 𝑥+1 =1 𝑥 2 +9𝑥−12= 2𝑥−3 𝑥+1 𝑥 2 +9𝑥−12=2 𝑥 2 −𝑥−3 𝑥 2 −10𝑥+9=0 𝑥−1 𝑥−9 =0 𝒙=𝟏 𝒐𝒓 𝒙=𝟗 ? ?

Test Your Understanding 𝑥 3 + 10 𝑥−1 =4 𝑥 2 −13𝑥+42=0 𝑥−6 𝑥−7 =0 𝑥=6 𝑜𝑟 𝑥=7 Solve, giving your answer to 3sf. 1 𝑥 +5= 2 𝑥 2 𝑥+5 𝑥 2 =2 5 𝑥 2 +𝑥−2 𝑥=−0.740 𝑜𝑟 𝑥=0.540 ? ? 4𝑥−1 5 + 𝑥+4 2 =3 𝒙= 𝟏𝟐 𝟏𝟑 ? 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎

Exercise 3 Give exact answers unless otherwise specified. 1 𝑥 +3= 2 𝑥 2 𝒙=−𝟏, 𝟐 𝟑 ℎ+7 3 + 2ℎ−1 2 = 5 6 𝒉=−𝟎.𝟕𝟓 Give your answer to 3sf: 2 𝑦 2 + 9 𝑦 −7=0 𝒚=−𝟎.𝟏𝟗𝟑 𝒐𝒓 𝟏.𝟒𝟖 𝑥 2 − 2 𝑥+1 =1 𝒙=𝟑, −𝟐 5 2𝑥+1 2 4𝑥+5 =5𝑥−1 𝒙=𝟏𝟎 ? Find exact solutions to: 𝑥+ 3 𝑥 =7 𝒙= 𝟕± 𝟑𝟕 𝟐 𝑥 3 − 4 𝑥−5 =2 𝒙=𝟐, 𝟗 𝑥−2 5 − 6 𝑥 =1 𝒙=−𝟑, 𝟏𝟎 1 ? 6 ? 2 ? 7 3 ? ? 8 ? 4 5 ?