Sec 3 E-Math Quadratic equations worksheet with answers

A free Sec 3 E-Math worksheet on quadratic equations: 8 questions, 30 marks, short questions first and then structured ones, in the school exam style. The answers and full worked solutions are on the last page.

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Sec 3 E-Math worksheet: Quadratic equations

Total: 30 marksSuggested time: 45 minutes

Name: ______________________________Class: ____________ Date: ____________

Answer all questions. Show all your working. Give non-exact answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.

Section A

The curve

y = ax^2 + bx + c

passes through the points

(0, -4)

and

(-2, -20)

. Its line of symmetry is

x = 1

. Find the values of

a

b

and

c

, and write down the equation of the curve.

Answer[3]

Solve

x^{2} - 9x + 7 = 0

, giving your answers correct to 2 decimal places.

Answer[3]

(a)Solve

6x^{2} - 7x - 10 = 0

Answer (a)[2]

(b)Solve

(x - 5)(x - 3) = 3

Answer (b)[2]

Section B

Solve the equation

x^{2} -8x = -9

, giving your answers correct to 2 decimal places.

Answer[3]

(a)Factorise

2x^{2} - 15x + 28

Answer (a)[2]

(b)Hence solve the equation

2x^{2} - 15x + 28 = 0

Answer (b)[1]

(a)Express

x^{2} + 2x - 4

in the form

(x + h)^2 + k

, where

h

and

k

are constants.

Answer (a)[2]

(b)Hence solve the equation

x^{2} + 2x - 4 = 0

, giving your answers correct to 2 decimal places.

Answer (b)[2]

Solve the equation

\dfrac{5}{x - 5} + \dfrac{3}{x + 5} = 3

, giving your answers correct to 2 decimal places.

Answer[4]

The diagram shows a right-angled triangle with sides

x

cm,

(x + 7)

cm and

(x + 8)

cm.

A right-angled triangle with legs x cm and (x + 7) cm and hypotenuse (x + 8) cm

(a)Show that

x^2 - 2x - 15 = 0

[3]

(b)Solve the equation

x^2 - 2x - 15 = 0

and find the value of

x

Answer (b)[2]

(c)Find the area of the triangle.

Answer (c)cm²[1]

END OF PAPER

Answers and worked solutions

Question 1

Answer:

y = -2x^{2} + 4x - 4

[3]

c = -4

Mirror image:

(2, -4)

-4 = a(2)^2 + b(2) + -4

0 = 4a + 2b

b = -2a

-20 = a(-2)^2 + (-2a)(-2) + -4

8a = -16

a = -2

b = 4

y = -2x^{2} + 4x - 4

Question 2

Answer:

x = 8.14

x = 0.86

(2 d.p.)[3]

a = 1,\ b = -9,\ c = 7

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

x = \dfrac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(7)}}{2(1)}

x = \dfrac{9 \pm \sqrt{53}}{2}

x = 8.1400\ldots

x = 0.8599\ldots

x = 8.14

x = 0.86

(2 d.p.)

Question 3

(a)Answer:

x = -\dfrac{5}{6}

x = 2

[2]

(6x + 5)(x - 2) = 0

6x + 5 = 0

x - 2 = 0

x = -\dfrac{5}{6}

x = 2

(b)Answer:

x = 6

x = 2

[2]

x^{2} - 8x + 15 = 3

x^{2} - 8x + 12 = 0

(x - 6)(x - 2) = 0

x - 6 = 0

x - 2 = 0

x = 6

x = 2

Question 4

Answer:

x = 6.65

x = 1.35

[3]

x^{2} -8x = -9

x^{2} - 8x + 9 = 0

a = 1,\ b = -8,\ c = 9

x = \dfrac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(9)}}{2(1)}

x = \dfrac{8 \pm \sqrt{28}}{2}

x = 6.6457\ldots

x = 1.3542\ldots

x = 6.65

x = 1.35

(2 d.p.)

Question 5

(a)Answer:

(2x - 7)(x - 4)

[2]

2x^{2} - 15x + 28

= (2x - 7)(x - 4)

(b)Answer:

x = \dfrac{7}{2}

x = 4

[1]

(2x - 7)(x - 4) = 0

x = \dfrac{7}{2}

x = 4

Question 6

(a)Answer:

(x + 1)^2 - 5

[2]

Half the coefficient of

x

1

x^{2} + 2x - 4 = \left(x + 1\right)^2 - \left(1\right)^2 -4

= (x + 1)^2 - 5

(b)Answer:

x = 1.24

x = -3.24

[2]

(x + 1)^2 = 5

x = -1 \pm \sqrt{5}

x = 1.2360\ldots

x = -3.2360\ldots

x = 1.24

x = -3.24

(2 d.p.)

Question 7

Answer:

x = 6.82

x = -4.15

[4]

\dfrac{5}{x - 5} + \dfrac{3}{x + 5} = 3

5(x + 5) + 3(x - 5) = 3(x - 5)(x + 5)

3x^{2} - 8x - 85 = 0

x = \dfrac{-(-8) \pm \sqrt{(-8)^2 - 4(3)(-85)}}{2(3)}

x = \dfrac{8 \pm \sqrt{1084}}{6}

x = 6.8206\ldots

x = -4.1540\ldots

x = 6.82

x = -4.15

(2 d.p.)

Question 8

(a)[3]

x^2 + (x + 7)^2 = (x + 8)^2

x^2 + x^2 + 14x + 49 = x^2 + 16x + 64

x^2 - 2x - 15 = 0

(shown)

(b)Answer:

x = 5

[2]

(x - 5)(x + 3) = 0

x = 5

x = -3

x = 5

(

x = -3

is rejected: a length can't be negative)

(c)Answer:

30

cm²[1]

The sides are

5

cm,

12

cm and

13

cm.Area

= \dfrac{1}{2} \times 5 \times 12

= 30

cm²

Practice questions by skill

Sec 3 E-Math Quadratic equations worksheet with answers

In the print window, choose Save as PDF to download it.

Need the formulas first? Read the Quadratic equations notes. Or practise it online with step-by-step help when you get one wrong.

Answers and worked solutions

Question 1

Answer:

y = -2x^{2} + 4x - 4

[3]

c = -4

Mirror image:

(2, -4)

-4 = a(2)^2 + b(2) + -4

0 = 4a + 2b

b = -2a

-20 = a(-2)^2 + (-2a)(-2) + -4

8a = -16

a = -2

b = 4

y = -2x^{2} + 4x - 4

Question 2

Answer:

x = 8.14

x = 0.86

(2 d.p.)[3]

a = 1,\ b = -9,\ c = 7

x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}

x = \dfrac{-(-9) \pm \sqrt{(-9)^2 - 4(1)(7)}}{2(1)}

x = \dfrac{9 \pm \sqrt{53}}{2}

x = 8.1400\ldots

x = 0.8599\ldots

x = 8.14

x = 0.86

(2 d.p.)

Question 3

(a)Answer:

x = -\dfrac{5}{6}

x = 2

[2]

(6x + 5)(x - 2) = 0

6x + 5 = 0

x - 2 = 0

x = -\dfrac{5}{6}

x = 2

(b)Answer:

x = 6

x = 2

[2]

x^{2} - 8x + 15 = 3

x^{2} - 8x + 12 = 0

(x - 6)(x - 2) = 0

x - 6 = 0

x - 2 = 0

x = 6

x = 2

Question 4

Answer:

x = 6.65

x = 1.35

[3]

x^{2} -8x = -9

x^{2} - 8x + 9 = 0

a = 1,\ b = -8,\ c = 9

x = \dfrac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(9)}}{2(1)}

x = \dfrac{8 \pm \sqrt{28}}{2}

x = 6.6457\ldots

x = 1.3542\ldots

x = 6.65

x = 1.35

(2 d.p.)

Question 5

(a)Answer:

(2x - 7)(x - 4)

[2]

2x^{2} - 15x + 28

= (2x - 7)(x - 4)

(b)Answer:

x = \dfrac{7}{2}

x = 4

[1]

(2x - 7)(x - 4) = 0

x = \dfrac{7}{2}

x = 4

Question 6

(a)Answer:

(x + 1)^2 - 5

[2]

Half the coefficient of

x

1

x^{2} + 2x - 4 = \left(x + 1\right)^2 - \left(1\right)^2 -4

= (x + 1)^2 - 5

(b)Answer:

x = 1.24

x = -3.24

[2]

(x + 1)^2 = 5

x = -1 \pm \sqrt{5}

x = 1.2360\ldots

x = -3.2360\ldots

x = 1.24

x = -3.24

(2 d.p.)

Question 7

Answer:

x = 6.82

x = -4.15

[4]

\dfrac{5}{x - 5} + \dfrac{3}{x + 5} = 3

5(x + 5) + 3(x - 5) = 3(x - 5)(x + 5)

3x^{2} - 8x - 85 = 0

x = \dfrac{-(-8) \pm \sqrt{(-8)^2 - 4(3)(-85)}}{2(3)}

x = \dfrac{8 \pm \sqrt{1084}}{6}

x = 6.8206\ldots

x = -4.1540\ldots

x = 6.82

x = -4.15

(2 d.p.)

Question 8

(a)[3]

x^2 + (x + 7)^2 = (x + 8)^2

x^2 + x^2 + 14x + 49 = x^2 + 16x + 64

x^2 - 2x - 15 = 0

(shown)

(b)Answer:

x = 5

[2]

(x - 5)(x + 3) = 0

x = 5

x = -3

x = 5

(

x = -3

is rejected: a length can't be negative)

(c)Answer:

30

cm²[1]

The sides are

5

cm,

12

cm and

13

cm.Area

= \dfrac{1}{2} \times 5 \times 12

= 30

cm²