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Linear and Quadratic Equations in Context

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen, paper and a calculator handy, to work out your answers.

Question 13 marks

Sam is x years old.

Sam's aunt is (3x+4) years old.

In 8 years' time, the ratio of Sam's age to his aunt's age will be 3:7

Work out the value of x.

Hint

Add 8 to both ages, then use the fact that a:b=3:7 means 7a=3b.

Worked solution
  1. In 8 years: Sam is x+8, his aunt is 3x+12
  2. (x+8):(3x+12)=3:7 so 7(x+8)=3(3x+12)
  3. 7x+56=9x+36
  4. 20=2x, so x=10
  5. Check: in 8 years the ages are 18 and 42, and 18:42=3:7
  6. Answer: x=10

Question 23 marks

A right-angled triangle has sides of length x cm, (x+1) cm and (x+9) cm.

The longest side is (x+9) cm.

Work out the perimeter of the triangle.

Hint

Use Pythagoras with (x+9) as the hypotenuse to get a quadratic, then reject the solution that gives a negative length.

Worked solution
  1. Pythagoras: x^2+(x+1)^2=(x+9)^2
  2. x^2+x^2+2x+1=x^2+18x+81
  3. x^2-16x-80=0
  4. (x-20)(x+4)=0, so x=20 or x=-4
  5. A length cannot be negative, so x=20
  6. Sides are 20, 21 and 29 cm
  7. Perimeter =20+21+29=70
  8. Answer: 70 cm

Question 3Challenge5 marks

A rectangle has length (2x+1) cm and width x cm.

A square has sides of length (x+3) cm.

The area of the rectangle is 25% more than the area of the square.

(a)

Which equation does x satisfy?

1 mark

(b)

Work out the value of x.

Give your answer to 3 significant figures.

Do not use trial and improvement.

3 marks

(c)

Work out the perimeter of the rectangle. Use your unrounded value of x.

Give your answer to 3 significant figures.

1 mark

Hint

25% more means multiply the square's area by 1.25; expand (x+3)^2 fully before collecting terms.

Worked solution

Part (a)

  1. Rectangle area =x(2x+1)=2x^2+x
  2. 25% more than the square means \times1.25: 2x^2+x=1.25(x+3)^2
  3. 1.25(x^2+6x+9)=1.25x^2+7.5x+11.25
  4. 2x^2+x-1.25x^2-7.5x-11.25=0
  5. 0.75x^2-6.5x-11.25=0; multiply by 4
  6. Answer: 3x^2-26x-45=0

Part (b)

  1. Use the formula with a=3, b=-26, c=-45
  2. x=\dfrac{26\pm\sqrt{(-26)^2-4\times3\times(-45)}}{2\times3}=\dfrac{26\pm\sqrt{1216}}{6}
  3. x=10.145\ldots or x=-1.478\ldots
  4. x is a length, so it must be positive
  5. Answer: x=10.1 (3 s.f.)

Part (c)

  1. Perimeter =2(2x+1)+2x=6x+2
  2. Use the unrounded value x=10.145\ldots
  3. 6\times10.145\ldots+2=62.87\ldots
  4. Answer: 62.9 cm (3 s.f.)

More on this topic: Linear and Quadratic Equations in Context worksheet with full solutions

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