Algebraic Fractions - GCSE Revision Pack Overview
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Explore algebraic fractions through GCSE revision materials including simplification, addition, subtraction, and solving equations involving algebraic fractions leading to quadratic equations. Practice exercises and tips are provided for enhanced understanding.
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GCSE: Algebraic Fractions Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com GCSE Revision Pack Ref: 53ii, 53iii, 92 Last modified: 31st August 2015
GCSE Specification (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.
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 Fail Win! 2 = 3? + 2 2 ? = 2? + 3 Fail Win!
Starter (Click your answer) Are these algebraic steps correct? ?2? ? + ? ?? ? Fail Win!
Starter To cancel or not to cancel, that is the question? (Click your answer) y2 + x 2 + x s(4 + z) s Fail ?2+ 2 = ? + 2 Fail Win! Win! Fail Win! 1 + r 2 pq(r+2) + 1 pq (2x+1)(x 2) x 2 Fail - 1 Win! Fail Win! Fail Win!
What did we learn? Bro Tip #1: You can t add or subtract a term which is trapped inside a bracket, fraction or root. 2 ? ? = 2? + 1 2 ? = 3? + 1 Bro 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!)
Simplifying Algebraic Fractions Bro Tip: Just factorise top and bottom, then cancel! 2?2+ 4? ?2 4 2? ? = ? 2 3? + 3 ?2+ 3? + 2= 3 ? ? + 2 2?2 5? 3 6?3 2?4 = 2? + 1 ? 2?3
Test Your Understanding Bro Tip: Sometimes they ve done part of the factorising for you! ?(? + 1) ?2 1 2?2+ 5? 3 ?2 9 ? ? ? 1 2? 1 ? 3 ? ?2+ 2?? + ?2 2? + 2? ? + ? ? 2
Negating a difference 4 ? = ? 4 2? 9 = 9 2? ? ? 1 ? ? 1= 1 ? 3 2? 2 ? 2? 3 ? + 1 =? 2 ? + 1 ?
Exercise 1 ? + 105 ? + 104= ? + 10 2? + 6 2? =? + 3 ? 1 8 ? ? ?2+ 2? + 1 ?2+ 3? + 2=? + 1 4? + 8 3? + 6=4 9 ? ? 2 3 ? + 2 ?2+ ? 6 ?2 7? + 10=? + 3 ? 5 ? ?2 8? + 15 2?2 7? 15= 3 10 ? 3 2? + 3 ? 2? + 10 ?2 25= 2 ? 4 ? 5 ?2 9 ? + 3 2? 1 ? 11 2?2 7? + 3= ?2 9 2? + 6=? 3 5 ? 2 6?2 ? 1 4?2 1 =3? + 1 2? + 1 12 ?2+ ? 2 ?2 4 =? 1 ? 2 ? ? 6 6?2+ 3? 4?2 1= 2?2+ 4? 3?2+ 7? + 2 9?2 1 3? ? 7 ? 13 3?2 ?= 2 2? 1
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 ? 1 4 3 ?=3? 4 ? 1 ? ? 1 =3? 4? + 4 ? ? 1 ? + 4 ? ? 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 3 5? + 1 ? + 1 ? 1 ? ? + 1+ ? 1= ? ? ? 2 ?2+ 4? ? 2 ? + 1 2 ? ? ? 2 ? + 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 ?+1 ?? ? ?=? + ? 2 1 ? 15 ? + 3 ? 6 ? 1 ? + 3 ? 6= 8 1 2 3? + 4 ? + 4 ? 4 ? 2 1 ? 1 ? 9 ? + 4+ ? 4= 2 ?2 9+ ? + 3= ? + 3 ? 3 ? + 3 4 +? 5 =7? 11 12 2 4 2? 10 ? 3 ? 2 ? 4 ?= 3 2 ? 4 ? 2 1 ? 12 ? + 4 ? 4 ? 3 4 3? + 7 ? + 12 ? 11 ? + 4 ? 4 = ? + 1+ ? + 12= 4 2 1 ? + 3 ? 1 2 ? + 5 ? 12 5 ? 1 ? + 1= ? 1 ? + 1 ? 3 3? 1= ? 3 3? 1 1 4 3 2? + 7 ? 3?2+ 6? + 2 ?(? + 1)(? + 2) ? 1 ?+ 1 1 6 2? 1 2? + 1= ? + 1+ ? + 2 = 2? 1 2? + 1 ?=2? ?2 1 ? ? + 1 ? + 1 ? 2 1 1 ? + 2 ? + 110 ? ? 7 2 ? + 110+ ? + 19 =
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 ?2 turns equation into a quadratic, this is simpler. 8 ?2+4 8 + 4? = 4?2 2 + ? = ?2 ?2 ? 2 = 0 ? + 1 ? 2 = 0 ? = ? ?? ? = ? ? 4 ?= 4 2? 3+ ?2+ 9? 12 2? 3 ? + 1 ? + 1= 1 = 1 ? ?2+ 9? 12 = 2? 3 ? + 1 ?2+ 9? 12 = 2?2 ? 3 ?2 10? + 9 = 0 ? 1 ? 9 = 0 ? = ? ?? ? = ? ?
Test Your Understanding ? 3+ 10 ? 1= 4 Solve, giving your answer to 3sf. ?2 13? + 42 = 0 ? 6 ? 7 = 0 ? = 6 ?? ? = 7 ? 1 ?+ 5 = 2 ?2 ? 4? 1 5 +? + 4 ? + 5?2= 2 5?2+ ? 2 ? = 0.740 ?? ? = 0.540 = 3 2 ? =?? ?? ? ? = ? ?2 4?? 2?
Exercise 3 Give exact answers unless otherwise specified. 1 ?+ 3 = 2 ?2 ? = ?,? ? Find exact solutions to: ? +3 ?= 7 1 ? ? =? ?? ? 6 ? + 7 3 +2 1 =5 ? = ?.?? ? 2 6 2 ? 3 4 ? ? 5= 2 ? = ?,? 7 Give your answer to 3sf: 2 ?2+9 3 ? 2 5 6 ? = ?,?? ? ? 7 = 0 ? = ?.??? ?? ?.?? ? 8 ?= 1 ? 2 2 ? ? + 1= 1 ? = ?, ? 4 5 2? + 12 4? + 5 5 ? = 5? 1 ? = ??
