Review Solutions — Quadratic Expressions & Equations

1. Factorise completely. Fluency

   • (a) x² + 7x + 12: Need two numbers that multiply to 12 and add to 7: use 3 and 4. (x + 3)(x + 4)
   • (b) x² − x − 20: Multiply to −20, add to −1: use −5 and 4. (x − 5)(x + 4)
   • (c) 2x² + 5x + 2: ac = 4, need sum 5: use 4 and 1. 2x²+4x+x+2 = 2x(x+2)+1(x+2) = (2x+1)(x+2)
   • (d) 4x² − 9: Difference of two squares: (2x)² − 3². (2x − 3)(2x + 3)

2. Expand and simplify. Fluency

   • (a) (x+3)(x−5): x²−5x+3x−15 = x²−2x−15
   • (b) (2x−1)²: 4x²−4x+1. 4x²−4x+1
   • (c) (x+4)(x−4): Difference of two squares. x²−16
   • (d) 3(x−2)(x+7): First expand (x−2)(x+7) = x²+5x−14. Multiply by 3: 3x²+15x−42

3. Solve by factorising. Fluency

   • (a) x²−8x+15=0: Multiply to 15, add to −8: −3 and −5. (x−3)(x−5)=0. x=3 or x=5
   • (b) x²+3x=0: HCF = x: x(x+3)=0. x=0 or x=−3
   • (c) 2x²−5x−3=0: ac=−6, sum −5: use −6 and 1. 2x²−6x+x−3=0 ⇒ 2x(x−3)+1(x−3)=0 ⇒ (2x+1)(x−3)=0. x=−1/2 or x=3
   • (d) x²=25: Rearrange: x²−25=0. Difference of two squares: (x−5)(x+5)=0. x=5 or x=−5

4. Quadratic formula — exact answers. Fluency

   • (a) x²−4x+1=0: a=1, b=−4, c=1. Δ=16−4=12. √12=2√3. x=(4±2√3)/2 = 2 ± √3
   • (b) 3x²+2x−2=0: a=3, b=2, c=−2. Δ=4+24=28. √28=2√7. x=(−2±2√7)/6 = (−1 ± √7)/3

5. Key features of y = −x² + 4x + 5. Fluency

   • (a) Direction and y-intercept: a = −1 < 0 ⇒ opens downward. y-intercept: (0, 5).
   • (b) Axis and vertex: Axis: x = −4/(2×−1) = x = 2. Vertex: y(2) = −4+8+5 = 9 ⇒ (2, 9).
   • (c) x-intercepts: Set y=0: −x²+4x+5=0 ⇒ x²−4x−5=0 ⇒ (x−5)(x+1)=0. x-intercepts: (5, 0) and (−1, 0).

6. Discriminant with parameter k. Understanding

   • (a) Discriminant: a=1, b=−6, c=k. Δ = 36 − 4k
   • (b) Exactly one solution: Δ=0 ⇒ 36−4k=0 ⇒ k=9
   • (c) Two distinct solutions: Δ>0 ⇒ 36−4k>0 ⇒ k<9
   • (d) Solve when k=5: x²−6x+5=0. Factorise: (x−1)(x−5)=0. x=1 or x=5

7. y = (x−2)²−9. Understanding

   • (a) Vertex and axis: Vertex: (2, −9). Axis of symmetry: x = 2.
   • (b) Standard form: Expand: (x−2)² = x²−4x+4. So y = x²−4x+4−9 = y = x²−4x−5.
   • (c) x-intercepts: Set y=0: x²−4x−5=0 ⇒ (x−5)(x+1)=0. x-intercepts: (5,0) and (−1,0).
   • (d) y-intercept: Set x=0: y = 0−0−5 = −5. Or from vertex form: (0−2)²−9=4−9=−5. y-intercept: (0,−5).

8. y = 2x²−8x+k through (0,6). Understanding

   • (a) Find k: Substitute (0,6): 6 = 2(0)−8(0)+k = k. k = 6. Equation: y = 2x²−8x+6.
   • (b) Vertex: x = −(−8)/(2×2) = 8/4 = 2. y(2) = 2(4)−16+6 = 8−16+6 = −2. Vertex: (2,−2).
   • (c) x-intercepts: 2x²−8x+6=0 ⇒ x²−4x+3=0 ⇒ (x−1)(x−3)=0. x-intercepts: (1,0) and (3,0).

9. Rectangle dimensions. Understanding

   • (a) Quadratic equation: Width = w, length = w+4. Area: w(w+4)=45. w²+4w−45=0
   • (b) Solve: Multiply to −45, add to 4: use 9 and −5. (w+9)(w−5)=0. w=−9 or w=5. Reject w=−9 (width cannot be negative). w=5 cm
   • (c) Dimensions: Width = 5 cm, Length = 9 cm. Check: 5×9=45 cm² ✓

10. Stone from a cliff. Problem Solving

    • (a) Initial height: h(0) = 0+0+35 = 35 m.
    • (b) Maximum height: Vertex: t = −2/(2×−1) = 1 s. h(1) = −1+2+35 = 36 m at t = 1 s.
    • (c) Hits the water: Set h=0: −t²+2t+35=0 ⇒ t²−2t−35=0 ⇒ (t−7)(t+5)=0. Reject t=−5. t = 7 s.

11. Consecutive even integers. Problem Solving

    • (a) Equation: Integers are n and n+2. n(n+2)=168 ⇒ n²+2n−168=0.
    • (b) Solve: Multiply to −168, add to 2: use 14 and −12. (n+14)(n−12)=0. n=12 or n=−14.
    • (c) All pairs: n=12 ⇒ integers are 12 and 14. n=−14 ⇒ integers are −14 and −12. Both pairs satisfy the condition: 12×14=168 ✓ and (−14)×(−12)=168 ✓.

12. Controlling solutions of x²+kx+9=0. Problem Solving

    • (a) Discriminant: a=1, b=k, c=9. Δ = k²−36
    • (b) Exactly one solution: Δ=0 ⇒ k²=36 ⇒ k=6 or k=−6
    • (c) Two distinct solutions: Δ>0 ⇒ k²>36 ⇒ k>6 or k<−6
    • (d) No real solutions: Δ<0 ⇒ k²<36 ⇒ −6 < k < 6