Mathbench

5. Quadratics

Solving quadratic equations by factorisation, by formula, and finding turning points. Set for OMPT-A, OMPT-B, OMPT-D, OMPT-E, OMPT-F and OMPT-G.

All OMPT worksheets

Questions

Sheet length
1

Solve x2+10x+24=0x^{2} + 10 x + 24 = 0 and give the LARGER root.

Answer:

2

Solve exactly and give the LARGER solution: 2x2−7x+4=02 x^{2} - 7 x + 4 = 0

Answer:

3

The parabola f(x)=−x2−8x−12f(x) = - x^{2} - 8 x - 12 has a maximum. At what value of xx does it occur?

Answer:

4

A parabola and a line meet where x2+2x=4x+8x^{2} + 2 x = 4 x + 8. Give the LARGER of the two xx-coordinates.

Answer:

5

Solve x2+3x−4=0x^{2} + 3 x - 4 = 0 and give the LARGER root.

Answer:

6

Solve exactly and give the LARGER solution: 2x2−4x−5=02 x^{2} - 4 x - 5 = 0

Answer:

7

The parabola f(x)=−4x2−8x−9f(x) = - 4 x^{2} - 8 x - 9 has a maximum. At what value of xx does it occur?

Answer:

8

A parabola and a line meet where x2+6x+18=−4x−6x^{2} + 6 x + 18 = - 4 x - 6. Give the LARGER of the two xx-coordinates.

Answer:

9

Solve x2+13x+42=0x^{2} + 13 x + 42 = 0 and give the LARGER root.

Answer:

10

Solve exactly and give the LARGER solution: x2+6x+7=0x^{2} + 6 x + 7 = 0

Answer:

11

The parabola f(x)=−x2+8x−16f(x) = - x^{2} + 8 x - 16 has a maximum. At what value of xx does it occur?

Answer:

12

A parabola and a line meet where x2−2x−17=−3x−5x^{2} - 2 x - 17 = - 3 x - 5. Give the LARGER of the two xx-coordinates.

Answer:

13

Solve x2+16x+63=0x^{2} + 16 x + 63 = 0 and give the LARGER root.

Answer:

14

Solve exactly and give the LARGER solution: x2−5x+5=0x^{2} - 5 x + 5 = 0

Answer:

15

The parabola f(x)=−4x2−8x−3f(x) = - 4 x^{2} - 8 x - 3 has a maximum. At what value of xx does it occur?

Answer:

16

A parabola and a line meet where x2−8x−8=−5x−4x^{2} - 8 x - 8 = - 5 x - 4. Give the LARGER of the two xx-coordinates.

Answer:

17

Solve x2+12x+35=0x^{2} + 12 x + 35 = 0 and give the LARGER root.

Answer:

18

Solve exactly and give the LARGER solution: x2+7x+3=0x^{2} + 7 x + 3 = 0

Answer:

19

The parabola f(x)=−4x2−24x−42f(x) = - 4 x^{2} - 24 x - 42 has a maximum. At what value of xx does it occur?

Answer:

20

A parabola and a line meet where x2−3x−32=−x−8x^{2} - 3 x - 32 = - x - 8. Give the LARGER of the two xx-coordinates.

Answer:

21

Solve x2−14x+45=0x^{2} - 14 x + 45 = 0 and give the LARGER root.

Answer:

22

Solve exactly and give the LARGER solution: 2x2+5x−8=02 x^{2} + 5 x - 8 = 0

Answer:

23

The parabola f(x)=−4x2−8x−2f(x) = - 4 x^{2} - 8 x - 2 has a maximum. At what value of xx does it occur?

Answer:

24

A parabola and a line meet where x2−6x−2=−5x−2x^{2} - 6 x - 2 = - 5 x - 2. Give the LARGER of the two xx-coordinates.

Answer:

25

Solve x2+4x−45=0x^{2} + 4 x - 45 = 0 and give the LARGER root.

Answer:

26

Solve exactly and give the LARGER solution: x2+4x−6=0x^{2} + 4 x - 6 = 0

Answer:

27

The parabola f(x)=−3x2−18x−24f(x) = - 3 x^{2} - 18 x - 24 has a maximum. At what value of xx does it occur?

Answer:

28

A parabola and a line meet where x2+5x−15=−9x^{2} + 5 x - 15 = -9. Give the LARGER of the two xx-coordinates.

Answer:

29

Solve x2+7x=0x^{2} + 7 x = 0 and give the LARGER root.

Answer:

30

Solve exactly and give the LARGER solution: 2x2−x−2=02 x^{2} - x - 2 = 0

Answer:

Answers

Every answer below was re-derived independently before this page was built.

1

Answer: −4-4

(x+4)(x+6)=0\left(x + 4\right) \left(x + 6\right) = 0, so x=−6x = -6 or x=−4x = -4. The larger is −4-4.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → −6-6
  • Read the roots off with the signs unchanged → 66
2

Answer: 174+74\frac{\sqrt{17}}{4} + \frac{7}{4}

The discriminant is b2−4ac=17b^2 - 4ac = 17, which is positive but not a perfect square, so the roots are irrational and the formula is needed: x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}. The larger root is 174+74\frac{\sqrt{17}}{4} + \frac{7}{4}.

Common mistakes, and the answer each one gives:

  • Used +b+b in the numerator instead of −b-b → −74+174- \frac{7}{4} + \frac{\sqrt{17}}{4}
  • Divided by aa instead of 2a2a → 172+72\frac{\sqrt{17}}{2} + \frac{7}{2}
  • Gave the smaller root → 74−174\frac{7}{4} - \frac{\sqrt{17}}{4}
3

Answer: −4-4

The turning point of ax2+bx+cax^2+bx+c sits at x=−b2ax = -\dfrac{b}{2a}. Here a=−1a = -1 and b=−8b = -8, so x=−−82(−1)=−4x = -\dfrac{-8}{2(-1)} = -4. (Completing the square gives −1(x+4)2+4-1\left(x + 4\right)^2 + 4, which shows the same thing.)

Common mistakes, and the answer each one gives:

  • Used b2a\dfrac{b}{2a} without the minus sign → 44
  • Gave the MAXIMUM VALUE instead of where it happens → 44
4

Answer: 44

Where the graphs meet, the two expressions are equal. Bring everything to one side: x2−2x−8=0x^{2} - 2 x - 8 = 0, which factorises as (x−4)(x+2)\left(x - 4\right) \left(x + 2\right). The roots are −2-2 and 44.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → −2-2
  • Read the roots straight off the brackets with the signs unchanged → 22
5

Answer: 11

(x−1)(x+4)=0\left(x - 1\right) \left(x + 4\right) = 0, so x=1x = 1 or x=−4x = -4. The larger is 11.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → −4-4
  • Read the roots off with the signs unchanged → 44
6

Answer: 1+1421 + \frac{\sqrt{14}}{2}

The discriminant is b2−4ac=56b^2 - 4ac = 56, which is positive but not a perfect square, so the roots are irrational and the formula is needed: x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}. The larger root is 1+1421 + \frac{\sqrt{14}}{2}.

Common mistakes, and the answer each one gives:

  • Used +b+b in the numerator instead of −b-b → −1+142-1 + \frac{\sqrt{14}}{2}
  • Divided by aa instead of 2a2a → 2+142 + \sqrt{14}
  • Gave the smaller root → 1−1421 - \frac{\sqrt{14}}{2}
7

Answer: −1-1

The turning point of ax2+bx+cax^2+bx+c sits at x=−b2ax = -\dfrac{b}{2a}. Here a=−4a = -4 and b=−8b = -8, so x=−−82(−4)=−1x = -\dfrac{-8}{2(-4)} = -1. (Completing the square gives −4(x+1)2−5-4\left(x + 1\right)^2 - 5, which shows the same thing.)

Common mistakes, and the answer each one gives:

  • Used b2a\dfrac{b}{2a} without the minus sign → 11
  • Gave the MAXIMUM VALUE instead of where it happens → −5-5
8

Answer: −4-4

Where the graphs meet, the two expressions are equal. Bring everything to one side: x2+10x+24=0x^{2} + 10 x + 24 = 0, which factorises as (x+4)(x+6)\left(x + 4\right) \left(x + 6\right). The roots are −6-6 and −4-4.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → −6-6
  • Read the roots straight off the brackets with the signs unchanged → 66
9

Answer: −6-6

(x+6)(x+7)=0\left(x + 6\right) \left(x + 7\right) = 0, so x=−7x = -7 or x=−6x = -6. The larger is −6-6.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → −7-7
  • Read the roots off with the signs unchanged → 77
10

Answer: −3+2-3 + \sqrt{2}

The discriminant is b2−4ac=8b^2 - 4ac = 8, which is positive but not a perfect square, so the roots are irrational and the formula is needed: x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}. The larger root is −3+2-3 + \sqrt{2}.

Common mistakes, and the answer each one gives:

  • Used +b+b in the numerator instead of −b-b → 2+3\sqrt{2} + 3
  • Divided by aa instead of 2a2a → −6+22-6 + 2 \sqrt{2}
  • Gave the smaller root → −3−2-3 - \sqrt{2}
11

Answer: 44

The turning point of ax2+bx+cax^2+bx+c sits at x=−b2ax = -\dfrac{b}{2a}. Here a=−1a = -1 and b=8b = 8, so x=−82(−1)=4x = -\dfrac{8}{2(-1)} = 4. (Completing the square gives −1(x−4)2-1\left(x - 4\right)^2, which shows the same thing.)

Common mistakes, and the answer each one gives:

  • Used b2a\dfrac{b}{2a} without the minus sign → −4-4
  • Gave the MAXIMUM VALUE instead of where it happens → 00
12

Answer: 33

Where the graphs meet, the two expressions are equal. Bring everything to one side: x2+x−12=0x^{2} + x - 12 = 0, which factorises as (x−3)(x+4)\left(x - 3\right) \left(x + 4\right). The roots are −4-4 and 33.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → −4-4
  • Read the roots straight off the brackets with the signs unchanged → 44
13

Answer: −7-7

(x+7)(x+9)=0\left(x + 7\right) \left(x + 9\right) = 0, so x=−7x = -7 or x=−9x = -9. The larger is −7-7.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → −9-9
  • Read the roots off with the signs unchanged → 99
14

Answer: 52+52\frac{\sqrt{5}}{2} + \frac{5}{2}

The discriminant is b2−4ac=5b^2 - 4ac = 5, which is positive but not a perfect square, so the roots are irrational and the formula is needed: x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}. The larger root is 52+52\frac{\sqrt{5}}{2} + \frac{5}{2}.

Common mistakes, and the answer each one gives:

  • Used +b+b in the numerator instead of −b-b → −52+52- \frac{5}{2} + \frac{\sqrt{5}}{2}
  • Divided by aa instead of 2a2a → 5+5\sqrt{5} + 5
  • Gave the smaller root → 52−52\frac{5}{2} - \frac{\sqrt{5}}{2}
15

Answer: −1-1

The turning point of ax2+bx+cax^2+bx+c sits at x=−b2ax = -\dfrac{b}{2a}. Here a=−4a = -4 and b=−8b = -8, so x=−−82(−4)=−1x = -\dfrac{-8}{2(-4)} = -1. (Completing the square gives −4(x+1)2+1-4\left(x + 1\right)^2 + 1, which shows the same thing.)

Common mistakes, and the answer each one gives:

  • Used b2a\dfrac{b}{2a} without the minus sign → 11
  • Gave the MAXIMUM VALUE instead of where it happens → 11
16

Answer: 44

Where the graphs meet, the two expressions are equal. Bring everything to one side: x2−3x−4=0x^{2} - 3 x - 4 = 0, which factorises as (x−4)(x+1)\left(x - 4\right) \left(x + 1\right). The roots are −1-1 and 44.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → −1-1
  • Read the roots straight off the brackets with the signs unchanged → 11
17

Answer: −5-5

(x+5)(x+7)=0\left(x + 5\right) \left(x + 7\right) = 0, so x=−5x = -5 or x=−7x = -7. The larger is −5-5.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → −7-7
  • Read the roots off with the signs unchanged → 77
18

Answer: −72+372- \frac{7}{2} + \frac{\sqrt{37}}{2}

The discriminant is b2−4ac=37b^2 - 4ac = 37, which is positive but not a perfect square, so the roots are irrational and the formula is needed: x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}. The larger root is −72+372- \frac{7}{2} + \frac{\sqrt{37}}{2}.

Common mistakes, and the answer each one gives:

  • Used +b+b in the numerator instead of −b-b → 372+72\frac{\sqrt{37}}{2} + \frac{7}{2}
  • Divided by aa instead of 2a2a → −7+37-7 + \sqrt{37}
  • Gave the smaller root → −72−372- \frac{7}{2} - \frac{\sqrt{37}}{2}
19

Answer: −3-3

The turning point of ax2+bx+cax^2+bx+c sits at x=−b2ax = -\dfrac{b}{2a}. Here a=−4a = -4 and b=−24b = -24, so x=−−242(−4)=−3x = -\dfrac{-24}{2(-4)} = -3. (Completing the square gives −4(x+3)2−6-4\left(x + 3\right)^2 - 6, which shows the same thing.)

Common mistakes, and the answer each one gives:

  • Used b2a\dfrac{b}{2a} without the minus sign → 33
  • Gave the MAXIMUM VALUE instead of where it happens → −6-6
20

Answer: 66

Where the graphs meet, the two expressions are equal. Bring everything to one side: x2−2x−24=0x^{2} - 2 x - 24 = 0, which factorises as (x−6)(x+4)\left(x - 6\right) \left(x + 4\right). The roots are −4-4 and 66.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → −4-4
  • Read the roots straight off the brackets with the signs unchanged → 44
21

Answer: 99

(x−9)(x−5)=0\left(x - 9\right) \left(x - 5\right) = 0, so x=5x = 5 or x=9x = 9. The larger is 99.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → 55
  • Read the roots off with the signs unchanged → −5-5
22

Answer: −54+894- \frac{5}{4} + \frac{\sqrt{89}}{4}

The discriminant is b2−4ac=89b^2 - 4ac = 89, which is positive but not a perfect square, so the roots are irrational and the formula is needed: x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}. The larger root is −54+894- \frac{5}{4} + \frac{\sqrt{89}}{4}.

Common mistakes, and the answer each one gives:

  • Used +b+b in the numerator instead of −b-b → 54+894\frac{5}{4} + \frac{\sqrt{89}}{4}
  • Divided by aa instead of 2a2a → −52+892- \frac{5}{2} + \frac{\sqrt{89}}{2}
  • Gave the smaller root → −894−54- \frac{\sqrt{89}}{4} - \frac{5}{4}
23

Answer: −1-1

The turning point of ax2+bx+cax^2+bx+c sits at x=−b2ax = -\dfrac{b}{2a}. Here a=−4a = -4 and b=−8b = -8, so x=−−82(−4)=−1x = -\dfrac{-8}{2(-4)} = -1. (Completing the square gives −4(x+1)2+2-4\left(x + 1\right)^2 + 2, which shows the same thing.)

Common mistakes, and the answer each one gives:

  • Used b2a\dfrac{b}{2a} without the minus sign → 11
  • Gave the MAXIMUM VALUE instead of where it happens → 22
24

Answer: 11

Where the graphs meet, the two expressions are equal. Bring everything to one side: x2−x=0x^{2} - x = 0, which factorises as x(x−1)x \left(x - 1\right). The roots are 00 and 11.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → 00
  • Read the roots straight off the brackets with the signs unchanged → 00
25

Answer: 55

(x−5)(x+9)=0\left(x - 5\right) \left(x + 9\right) = 0, so x=−9x = -9 or x=5x = 5. The larger is 55.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → −9-9
  • Read the roots off with the signs unchanged → 99
26

Answer: −2+10-2 + \sqrt{10}

The discriminant is b2−4ac=40b^2 - 4ac = 40, which is positive but not a perfect square, so the roots are irrational and the formula is needed: x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}. The larger root is −2+10-2 + \sqrt{10}.

Common mistakes, and the answer each one gives:

  • Used +b+b in the numerator instead of −b-b → 2+102 + \sqrt{10}
  • Divided by aa instead of 2a2a → −4+210-4 + 2 \sqrt{10}
  • Gave the smaller root → −10−2- \sqrt{10} - 2
27

Answer: −3-3

The turning point of ax2+bx+cax^2+bx+c sits at x=−b2ax = -\dfrac{b}{2a}. Here a=−3a = -3 and b=−18b = -18, so x=−−182(−3)=−3x = -\dfrac{-18}{2(-3)} = -3. (Completing the square gives −3(x+3)2+3-3\left(x + 3\right)^2 + 3, which shows the same thing.)

Common mistakes, and the answer each one gives:

  • Used b2a\dfrac{b}{2a} without the minus sign → 33
  • Gave the MAXIMUM VALUE instead of where it happens → 33
28

Answer: 11

Where the graphs meet, the two expressions are equal. Bring everything to one side: x2+5x−6=0x^{2} + 5 x - 6 = 0, which factorises as (x−1)(x+6)\left(x - 1\right) \left(x + 6\right). The roots are −6-6 and 11.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → −6-6
  • Read the roots straight off the brackets with the signs unchanged → 66
29

Answer: 00

x(x+7)=0x \left(x + 7\right) = 0, so x=0x = 0 or x=−7x = -7. The larger is 00.

Common mistakes, and the answer each one gives:

  • Gave the smaller root → −7-7
  • Read the roots off with the signs unchanged → 77
30

Answer: 14+174\frac{1}{4} + \frac{\sqrt{17}}{4}

The discriminant is b2−4ac=17b^2 - 4ac = 17, which is positive but not a perfect square, so the roots are irrational and the formula is needed: x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}. The larger root is 14+174\frac{1}{4} + \frac{\sqrt{17}}{4}.

Common mistakes, and the answer each one gives:

  • Used +b+b in the numerator instead of −b-b → −14+174- \frac{1}{4} + \frac{\sqrt{17}}{4}
  • Divided by aa instead of 2a2a → 12+172\frac{1}{2} + \frac{\sqrt{17}}{2}
  • Gave the smaller root → 14−174\frac{1}{4} - \frac{\sqrt{17}}{4}