Problems — Algebra | Mathematics Department

1

**Rectangle dimensions.** A rectangle has length $(x + 3)$ cm and width $(x + 1)$ cm. Its area is $35$ cm². (a) Form an equation in $x$ and expand the brackets. (b) Solve the equation to find $x$. (c) State the dimensions of the rectangle.

Espace de travail

2

**Coins.** Aoife has a mix of 50p and 20p coins. She has 15 coins in total, with a total value of £4.80. How many of each coin does she have?

Espace de travail

3

**Surds in geometry.** A right-angled triangle has legs of length $\sqrt{3}$ cm and $\sqrt{12}$ cm. (a) Find the hypotenuse, giving the exact answer in simplest surd form. (b) Find the area of the triangle. (c) Find the perimeter, giving an exact answer in simplest surd form.

Espace de travail

4

**Algebraic identities.** (a) Show that $(x+y)^2 - (x-y)^2 = 4xy$. (b) Hence, without a calculator, find the value of $103^2 - 97^2$. (c) Generalise: $a^2 - b^2 = (a+b)(a-b)$. Use this to find $215^2 - 185^2$.

Espace de travail

5

**Consecutive integers.** Three consecutive integers have a sum of 102. (a) Set up an equation using $n$ as the smallest integer. (b) Find the three integers. (c) Show that for any three consecutive integers, their sum is divisible by 3.

Espace de travail

6

**Mixed factorising.** Factorise each fully. (a) $4x^2 - 25$ (b) $x^3 - 9x$ (c) $2x^2 + 8x + 6$ (d) $x^2 + 4x + 4 - y^2$ *(tricky)*

Espace de travail

7

**Two unknown coefficients.** A quadratic $x^2 + bx + c$ has roots 2 and 7. (a) Use the factorised form to find $b$ and $c$. (b) Verify by substituting $x = 2$ into $x^2 + bx + c$.

Espace de travail

8

**Rationalising and simplifying.** (a) Simplify $\sqrt{50} + \sqrt{18} - \sqrt{8}$. (b) Rationalise $\dfrac{4}{\sqrt{2} + 1}$. (c) Hence calculate $(\sqrt{2} + 1)\left(\dfrac{4}{\sqrt{2}+1}\right)$ — verify your answer makes sense.

Espace de travail

9

**Triangle perimeter system.** Two sides of an isosceles triangle have length $x + 3$ and the third (the base) has length $2x - 1$. The perimeter is $20$ cm. (a) Form an equation in $x$. (b) Find $x$ and state the side lengths. (c) Could this triangle exist if the perimeter were instead 5 cm? Explain.

Espace de travail

10

**Solving a quadratic.** Solve $x^2 - 5x = 6$.

Espace de travail

11

**Algebraic system in context.** A youth-club hires a hall. The cost is a fixed fee plus an hourly rate. - 3 hours costs £45. - 5 hours costs £67. (a) Set up two equations using $f$ (fixed fee) and $r$ (hourly rate). (b) Solve to find $f$ and $r$. (c) Predict the cost of an 8-hour booking.

Espace de travail

12

**Surds and Pythagoras.** A square has diagonal length 8 cm. (a) Find the exact side length of the square, simplifying any surds. (b) Find the exact area of the square. (c) Find the perimeter, giving an exact answer.

Espace de travail