★ Topic Review — Surds and Quadratic Functions

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This review covers all four lessons: Simplifying Surds, Rationalising Denominators, Quadratic Functions — Graphs and Features, and Solving Quadratic Equations. Questions increase in difficulty.

1. Fluency — Simplifying surds

   1. (a) Simplify √72.
   2. (b) Simplify 3√8 − √50 + 2√18.
   3. (c) Expand and simplify (√5 + 2)(√5 − 3).
   4. (d) Simplify (3√2)².

2. Fluency — Rationalising denominators

   1. (a) Rationalise the denominator of 6/√3.
   2. (b) Rationalise the denominator of 4/(1 + √5).
   3. (c) Simplify (2 + √3)/(2 − √3). Express in the form a + b√3.

3. Fluency — Quadratic features

   1. (a) For y = x² − 6x + 8, find the axis of symmetry and turning point.
   2. (b) Find the x-intercepts of y = x² − 6x + 8 by factorising.
   3. (c) State the range of y = x² − 6x + 8.
   4. (d) Find the discriminant of 2x² − 3x + 5 and describe the nature of its roots.

4. Fluency — Solving quadratic equations

   1. (a) Solve x² − 5x − 14 = 0 by factorising.
   2. (b) Solve 3x² + 2x − 1 = 0 using the quadratic formula.
   3. (c) Solve x² + 4x − 7 = 0 by completing the square. Leave in exact form.

5. Understanding

   1. (a) Find the equation of the quadratic with vertex (2, −5) that passes through the point (0, 3). Write in standard form.
   2. (b) A parabola has x-intercepts at x = −3 and x = 1, and passes through (0, −6). Find the equation in the form y = ax² + bx + c.
   3. (c) For what values of k does kx² − 4x + 1 = 0 have two distinct real solutions?

6. Understanding

   1. (a) Show that the line y = x + 3 does not intersect the parabola y = x² + 5x + 7.
   2. (b) Find the value(s) of m such that y = mx + 1 is tangent to y = x² − 2x + 4.
   3. (c) If α and β are the roots of x² − 3x − 5 = 0, find α + β and αβ without solving. Then find α² + β².

7. Problem Solving

   Combined skills — surds and quadratics.
   1. (a) Solve x² − 2√3 x − 1 = 0 using the quadratic formula. Leave in simplest surd form.
   2. (b) The length of a rectangle is (1 + √5) cm and the width is (√5 − 1) cm. Find the area and show it is a rational number. Then find the exact length of the diagonal.
   3. (c) A ball is thrown upward from the ground with height h = −5t² + 10√5 t metres after t seconds. Find the exact time(s) when the ball is at height 5 m, giving your answer in simplified surd form.

8. Problem Solving

   Investigation — nature of roots and graph intersections.
   1. (a) A quadratic y = x² + bx + 9 has exactly one x-intercept. Find all possible values of b and state the coordinates of the turning point for each case.
   2. (b) The parabola y = 2x² − 8x + k is always above the x-axis. Find the range of values of k.
   3. (c) Rationalise and simplify √(7 + 4√3). (Hint: write 7 + 4√3 as a perfect square (a + b)².)

9. Understanding — Modelling with quadratics

   1. (a) A ball is thrown upward. Its height (metres) after t seconds is h(t) = −5t² + 20t + 1. Find the maximum height and the time at which it occurs.
   2. (b) Find the time(s) when the ball is at a height of 16 m.
   3. (c) When does the ball hit the ground? Give your answer to 2 decimal places.

10. Understanding — Simultaneous equations with a quadratic

    1. (a) Find the points of intersection of y = x² − 3 and y = 2x.
    2. (b) Find the values of c for which y = x + c is tangent to y = x² + 2x.
    3. (c) For what range of k does y = x² − 4x + k have no x-intercepts?

11. Problem Solving — Nested surd simplification

    Challenge.
    1. (a) Simplify fully: (2√3 + √2)(2√3 − √2).
    2. (b) Show that 1/(√n + √(n+1)) = √(n+1) − √n using rationalisation. Use this to evaluate the sum 1/(√1 + √2) + 1/(√2 + √3) + 1/(√3 + √4).
    3. (c) A right-angled triangle has hypotenuse (1 + √3) cm and one leg of (1 + √2) cm. Find the exact length of the other leg in simplified form.