Problem-solving
Quadratic Equations
Show all working. Partial marks are given for method.
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1**Garden path.** A square garden has side $x$ m. A path of uniform width $1$ m is laid around the outside. The total area of garden + path is $(x + 2)^2$ m². If the area of the path alone is $32$ m², find $x$.
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2**Three methods.** Solve $x^2 - 6x + 5 = 0$ three ways: (a) By factorising. (b) By completing the square. (c) Using the quadratic formula. Do all three methods give the same answers?
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3**Projectile.** A ball is thrown vertically upward from height $1.5$ m at $20$ m/s. Its height after $t$ seconds is $$h(t) = -5t^2 + 20t + 1.5.$$ (a) When is the ball at height $11.5$ m? (b) When does it hit the ground (1 d.p.)? (c) What is the maximum height reached?
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4**Find unknown coefficient.** The equation $x^2 + bx + 12 = 0$ has roots that differ by 1. (a) Write expressions for the sum and product of the roots in terms of $b$. (b) Use part (a) and the given condition to find $b$.
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5**Box without a lid.** A rectangular piece of card is 10 cm wide and 16 cm long. Equal squares of side $x$ cm are cut from each corner, and the sides folded up to make an open box. (a) Find expressions for the dimensions of the box. (b) Find the value of $x$ that gives a base area of $40$ cm². (c) State any restrictions on $x$.
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6**Solving by completing the square.** Solve $x^2 + 8x - 5 = 0$, giving exact answers.
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7**Discriminant analysis.** The equation $x^2 + kx + (k+3) = 0$ has (a) two distinct real roots — find the values of $k$. (b) one repeated real root — find $k$. (c) no real roots — find $k$.
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8**Two number puzzle.** Two positive numbers differ by 3, and their product is 70. Find them.
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9**Speed problem.** A train travels 240 km. If it had travelled 10 km/h faster, the journey would have taken 20 minutes less. Find the original speed.
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10**Number identities.** Show that $(n+2)^2 - n^2 = 4n + 4$ for any integer $n$. Hence, find two positive integers $n$ such that $(n+2)^2 - n^2 = 80$.
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11**Roots and coefficients.** The quadratic $x^2 + bx + c = 0$ has roots $\alpha = 4$ and $\beta = -3$. (a) Use Vieta's formulas to find $b$ and $c$. (b) Verify by substituting each root into the equation. (c) Construct a new quadratic with roots $2\alpha$ and $2\beta$.
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12**Vertex form.** Write $f(x) = x^2 - 6x + 13$ in completed-square form and hence (a) state the minimum value of $f$ and where it occurs; (b) explain why $f(x) = 0$ has no real solutions.
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