AQA GCSE Maths: Equations and Inequalities

A rectangle has length $(x + 4)$(x+4) cm and width $(x - 1)$(x−1) cm. Its area is $36 \text{ cm}^2$36 cm2.

(a) Show that $x^2 + 3x - 40 = 0$x2+3x−40=0. (2 marks)

(b) Solve the equation $x^2 + 3x - 40 = 0$x2+3x−40=0. (2 marks)

(c) Hence write down the dimensions of the rectangle. (1 mark)

Solve the simultaneous equations $4x + 3y = 18,$4x+3y=18, $5x - 2y = 11.$5x−2y=11. (4 marks)

The equation $x^3 - 5x - 9 = 0$x3−5x−9=0 has a single root close to $x = 2.7$x=2.7.

(a) Show that the equation can be rearranged into the iterative formula $x_{n+1} = \sqrt[3]{5x_n + 9}.$xn+1​=35xn​+9​. (1 mark)

(b) Starting with $x_0 = 2.7$x0​=2.7, use the iterative formula to find $x_1$x1​, $x_2$x2​ and $x_3$x3​, giving each to $4$4 decimal places. (3 marks)

(a) Solve the inequality $4x - 7 \geq 2x + 5$4x−7≥2x+5. (2 marks)

(b) $n$n is an integer such that $-2 < n \leq 3$−2<n≤3. Write down all the possible values of $n$n. (1 mark)

By completing the square, write $x^2 + 6x + 1$x2+6x+1 in the form $(x + a)^2 + b$(x+a)2+b, where $a$a and $b$b are integers. (2 marks)

Solve $5(2x - 1) = 35$5(2x−1)=35. (1 mark)