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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 10 · Quadratic Equations

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
Garden path. A square garden has side xx m. A path of uniform width 11 m is laid around the outside. The total area of garden + path is (x+2)2(x + 2)^2 m². If the area of the path alone is 3232 m², find xx.

Working space

2Problem 2 of 12
Three methods. Solve x2−6x+5=0x^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?

Working space

3Problem 3 of 12
Projectile. A ball is thrown vertically upward from height 1.51.5 m at 2020 m/s. Its height after tt seconds is

h(t)=−5t2+20t+1.5.h(t) = -5t^2 + 20t + 1.5.


(a) When is the ball at height 11.511.5 m?
(b) When does it hit the ground (1 d.p.)?
(c) What is the maximum height reached?

Working space

4Problem 4 of 12
Find unknown coefficient. The equation x2+bx+12=0x^2 + bx + 12 = 0 has roots that differ by 1.

(a) Write expressions for the sum and product of the roots in terms of bb.
(b) Use part (a) and the given condition to find bb.

Working space

5Problem 5 of 12
Box without a lid. A rectangular piece of card is 10 cm wide and 16 cm long. Equal squares of side xx 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 xx that gives a base area of 4040 cm².
(c) State any restrictions on xx.

Working space

6Problem 6 of 12
Solving by completing the square. Solve x2+8x−5=0x^2 + 8x - 5 = 0, giving exact answers.

Working space

7Problem 7 of 12
Discriminant analysis. The equation x2+kx+(k+3)=0x^2 + kx + (k+3) = 0 has

(a) two distinct real roots — find the values of kk.
(b) one repeated real root — find kk.
(c) no real roots — find kk.

Working space

8Problem 8 of 12
Two number puzzle. Two positive numbers differ by 3, and their product is 70. Find them.

Working space

9Problem 9 of 12
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.

Working space

10Problem 10 of 12
Number identities. Show that (n+2)2−n2=4n+4(n+2)^2 - n^2 = 4n + 4 for any integer nn.

Hence, find two positive integers nn such that (n+2)2−n2=80(n+2)^2 - n^2 = 80.

Working space

11Problem 11 of 12
Roots and coefficients. The quadratic x2+bx+c=0x^2 + bx + c = 0 has roots α=4\alpha = 4 and β=−3\beta = -3.

(a) Use Vieta's formulas to find bb and cc.
(b) Verify by substituting each root into the equation.
(c) Construct a new quadratic with roots 2α2\alpha and 2β2\beta.

Working space

12Problem 12 of 12
Vertex form. Write f(x)=x2−6x+13f(x) = x^2 - 6x + 13 in completed-square form and hence

(a) state the minimum value of ff and where it occurs;
(b) explain why f(x)=0f(x) = 0 has no real solutions.

Working space