Quadratic Equations

14 Grade 10 Essential Mathematics questions on Quadratic Equations, with worked marking schemes. 14 questions are available on this topic in full.

1. Question 1

   4 marks

   Outline the steps you would follow to solve the equation $x^2-8x=20$ by completing the square, and state the solutions.

   (4 marks)

   Marking scheme

   Award: half the coefficient of x is −4, so add $(-4{)}^2=16$ to both sides (1). Left side written as a perfect square: $(x-4{)}^2=36$ (1). Square root taken on both sides with both signs: x − 4 = ±6 (1). Solutions x = 10 and x = −2 (1).

2. Question 2

   1 mark

   The equation $x^2+4x+7=0$ has:

   A. two different real solutions

   B. no real solutions

   C. infinitely many solutions

   D. exactly one real solution

   (1 mark)

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3. Question 3

   1 mark

   A rectangular kitchen garden in Kisii is 3 m longer than it is wide and covers an area of 40 m². If the width is w metres, which equation represents this information?

   A. $w^2-3w-40=0$

   B. $w^2+3w+40=0$

   C. $w^2+3=40$

   D. $w^2+3w-40=0$

   (1 mark)

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4. Question 4

   4 marks

   A rectangular vegetable plot at a home in Machakos is 4 m longer than it is wide and covers an area of 96 m². Taking the width to be w metres, form a quadratic equation and use it to find the dimensions of the plot.

   (4 marks)

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5. Question 5

   1 mark

   During a games lesson a ball is kicked vertically upwards and its height, h metres, after t seconds is given by $h=20t-5t^2$. The ball returns to the ground after:

   A. 2 seconds

   B. 4 seconds

   C. 5 seconds

   D. 20 seconds

   (1 mark)

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6. Question 6

   1 mark

   The solutions of the equation $x^2+2x-15=0$ are:

   A. x = −3 or x = −5

   B. x = 3 or x = 5

   C. x = −3 or x = 5

   D. x = 3 or x = −5

   (1 mark)

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7. Question 7

   3 marks

   A tomato farmer finds that her profit, P (in thousands of shillings), from selling x crates at the market is given by $P=30x-x^2$. Determine the values of x for which the profit is zero and explain what the larger value means to the farmer.

   (3 marks)

   Marking scheme

   Award: factorisation x(30 − x) = 0 (1). Values x = 0 and x = 30 (1). Explanation — at 30 crates the profit has fallen back to zero, so selling beyond that point (extra transport and handling costs, and the price dropping as the market is flooded) wipes out her gain (1).

8. Question 8

   1 mark

   A shopkeeper in Nyeri packs eggs in two trays whose numbers of eggs are consecutive whole numbers. The product of the two numbers is 156. The smaller number is:

   A. 78

   B. 12

   C. 11

   D. 13

   (1 mark)

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9. Question 9

   6 marks

   A rectangular fish pond measuring 10 m by 6 m is surrounded by a path of uniform width x metres. The pond and the path together cover an area of 96 m². Form a quadratic equation in x and hence determine the width of the path.

   (6 marks)

   Marking scheme

   Award: outer dimensions written as ($10+2x$) and ($6+2x$) (1). Equation ($10+2x$)$(6+2x)=96$ formed (1). Correct expansion $4x^2+32x+60=96$ (1). Simplified to $x^2+8x-9=0$ (1). Factorised (x + 9)(x − 1) = 0 giving x = −9 or x = 1 (1). x = −9 rejected as a width cannot be negative; path width = 1 m (1). Accept a check such as $12\times 8=96m^2$ in place of the rejection statement.

10. Question 10

    5 marks

    Solve the equation $3x^2+5x-2=0$ using the quadratic formula, showing each step of your working.

    (5 marks)

    Marking scheme

    Award: a = 3, b = 5, c = −2 correctly identified (1). Correct substitution into x = [−b ± $\sqrt{b^2-4ac}$] / 2a (1). Discriminant evaluated as 25 + 24 = 49 (1). $\sqrt{49}=7$ and the two branches (−5 + 7)/6 and (−5 − 7)/6 formed (1). Solutions x = $\frac{1}{3}$ and x = −2 (1).

11. Question 11

    1 mark

    To solve the equation $x^2-10x=24$ by completing the square, the number that must be added to both sides is:

    A. 5

    B. −5

    C. 100

    D. 25

    (1 mark)

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12. Question 12

    1 mark

    The solutions of the equation $2x^2-5x-3=0$ are:

    A. x = −3 or x = $\frac{1}{2}$

    B. x = 3 or x = −$\frac{1}{2}$

    C. x = 3 or x = $\frac{1}{2}$

    D. x = −3 or x = −$\frac{1}{2}$

    (1 mark)

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13. Question 13

    2 marks

    State the number of real solutions of the equation $x^2+6x+9=0$ and give a reason for your answer.

    (2 marks)

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14. Question 14

    3 marks

    Solve the equation $x^2-11x+30=0$ by factorisation.

    (3 marks)

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