Topic Review — Quadratic Expressions & Equations

Mixed practice covering all four lessons: expanding & factorising, solving by factorising, the quadratic formula, and graphing parabolas.

1. Factorise each expression completely. Fluency

   1. x² + 7x + 12
   2. x² − x − 20
   3. 2x² + 5x + 2
   4. 4x² − 9

2. Expand and simplify. Fluency

   1. (x + 3)(x − 5)
   2. (2x − 1)²
   3. (x + 4)(x − 4)
   4. 3(x − 2)(x + 7)

3. Solve each equation by factorising. Fluency

   1. x² − 8x + 15 = 0
   2. x² + 3x = 0
   3. 2x² − 5x − 3 = 0
   4. x² = 25

4. Apply the quadratic formula. Give exact answers in simplified form. Fluency

   1. x² − 4x + 1 = 0
   2. 3x² + 2x − 2 = 0

5. For the parabola y = −x² + 4x + 5, find: Fluency

   1. The direction it opens and the y-intercept.
   2. The axis of symmetry and vertex.
   3. The x-intercepts.

6. The equation x² − 6x + k = 0 contains the parameter k. Understanding

   1. Write an expression for the discriminant Δ in terms of k.
   2. Find the value of k for which the equation has exactly one solution.
   3. State the values of k for which there are two distinct real solutions.
   4. When k = 5, solve the equation by factorising.

7. The parabola y = (x − 2)² − 9. Understanding

   1. State the vertex and the axis of symmetry.
   2. Convert to standard form y = ax² + bx + c.
   3. Find the x-intercepts.
   4. State the y-intercept.

8. A parabola with equation y = 2x² − 8x + k passes through the point (0, 6). Understanding

   1. Find the value of k.
   2. Find the vertex of this parabola.
   3. Find the x-intercepts.

9. Rectangle dimensions. Understanding

   Geometry. A rectangle has an area of 45 cm². Its length is 4 cm more than its width.
   1. Let the width be w cm. Write a quadratic equation for the area.
   2. Solve the equation by factorising. Reject any solution that is not physically valid.
   3. State the dimensions of the rectangle.

10. Stone thrown from a cliff. Problem Solving

    Physics. A stone is thrown upward from the top of a cliff. Its height above the water (in metres) after t seconds is h = −t² + 2t + 35.
    1. What is the height of the cliff (the initial height at t = 0)?
    2. Find the maximum height reached by the stone and the time at which it occurs.
    3. Solve h = 0 to find when the stone hits the water. Reject any invalid solution.

11. Consecutive even integers. Problem Solving

    Number. The product of two consecutive even integers is 168.
    1. Let the smaller integer be n. Write a quadratic equation in standard form.
    2. Solve the equation by factorising.
    3. State all pairs of consecutive even integers that satisfy the condition.

12. Controlling the number of solutions. Problem Solving

    Algebraic analysis. Consider the equation x² + kx + 9 = 0.
    1. Write an expression for the discriminant Δ in terms of k.
    2. Find all values of k for which the equation has exactly one solution.
    3. Find the values of k for which the equation has two distinct real solutions.
    4. Find the values of k for which there are no real solutions.