Free Year 9 Expand a binomial product Practice | Skillo

Question 1 — Easy

Priya is designing a rectangular garden bed at her school. She models the area of the garden as the binomial product (x + 5)(x − 3), where x represents a length in metres. When she expands this expression, which monic quadratic expression is the result?

Answer: Expanding (x + 5)(x − 3) using the distributive law: x·x + x·(−3) + 5·x + 5·(−3) = x² − 3x + 5x − 15 = x² + 2x − 15. Option B incorrectly computes the sum of the linear terms as −3x + 5x = −2x instead of +2x. Option C uses 5 + 3 = 8 as the coefficient of x instead of computing 5 + (−3) = 2. Option D correctly gets the coefficient of x but incorrectly makes the constant term +15 instead of −15.

Question 2 — Medium

Kofi is writing a program to calculate the area of rectangular paddocks on an Australian farm. He has the quadratic expression x² − 7x + 10, where x is a positive integer representing a side length in metres. To find the possible dimensions of the paddock, Kofi needs to fully factorise this monic quadratic expression. Which factorised form is correct?

Answer: To factorise x² − 7x + 10, find two integers whose product is +10 and whose sum is −7. These are −2 and −5, since (−2) × (−5) = 10 and (−2) + (−5) = −7. This gives (x − 2)(x − 5). Option A gives a product of −10 and a sum of −11 when expanded, so it is incorrect. Option B gives a product of −10 and a sum of −3, which is also incorrect. Option D gives a product of −10 and a sum of +3, so it is incorrect.

Question 3 — Hard

Mei is tiling a square courtyard at the Canberra community centre. She models the area of the courtyard as x² + 6x + 9. She claims this quadratic expression can be written as a perfect square binomial product. Which of the following is the fully factorised form of x² + 6x + 9?

Answer: x² + 6x + 9 is a perfect square trinomial because 9 = 3² and 6 = 2 × 3, so it factorises as (x + 3)². Expanding (x + 3)² confirms: x² + 3x + 3x + 9 = x² + 6x + 9. Option B expands to x² + 10x + 9, which does not match. Option C expands to x² + 9x + 18, which does not match. Option D is the difference of squares identity and expands to x² − 9, which does not match.