Solving quadratic equationsAQA A-Level Maths: Subtopic test

1. 1
   The quadratic equation $2x^{2}-5x-12=0$.

   (a)

   Find the solutions of the equation.

   [1 mark]

   • A$x=4$ or $x=-\frac32$
   • B$x=-4$ or $x=\frac32$
   • C$x=\frac32$ or $x=4$
   • D$x=8$ or $x=-3$

   (b)

   Find the sum of the two solutions.

   [1 mark]

   • A$-\frac52$
   • B$-6$
   • C$5$
   • D$\frac52$

   (c)

   Hence, or otherwise, solve $2(y+1)^2-5(y+1)-12=0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The equation $x^{4}-13x^{2}+36=0$.

   (a)

   The substitution $u=x^{2}$ is made. Which equation in $u$ results?

   [1 mark]

   • A$u^2-13u-36=0$
   • B$u^4-13u^2+36=0$
   • C$u^2-13u+36=0$
   • D$u^2-13\sqrt{u}+36=0$

   (b)

   How many real solutions does the equation have for $x$?

   [1 mark]

   • A$2$
   • B$4$
   • C$3$
   • D$8$

   (c)

   Solve the equation.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The equation $x-7\sqrt{x}+10=0$, where $x\geq0$.

   (a)

   Show that, with $u=\sqrt{x}$, the equation becomes $u^2-7u+10=0$, and hence find the values of $x$.

   [3 marks]

   (b)

   Hence solve $\left(y^{2}+1\right)-7\sqrt{y^{2}+1}+10=0$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A rectangular garden has length $(x+5)$ metres and width $(x-1)$ metres, where $x>1$.

   (a)

   The area of the garden is 72 m$^2$.
   (i) Show that $x^2+4x-77=0$.
   (ii) Solve this equation and hence find the length and width of the garden.

   [6 marks]

   (b)

   The garden has a diagonal of length 15 m. Find the length and width of the garden, giving your answers to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions