Solving Quadratic Equations

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Solving Quadratic Equations — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionSolving Using the Quadratic FormulaAssessment Practice

5 marks~8 minCriterion A

A quadratic function is defined by

f(x) = 2x^2 + bx + 18

, where

b

is a constant.

The diagram shows the parabola

y = f(x)

crossing the

x

-axis at

x = \dfrac{3}{2}

and at a second point labelled

p

Write down the

y

-intercept of

y = f(x)

. [1]

Given that

x = \dfrac{3}{2}

is a root of

f(x) = 0

, calculate the value of

b

and hence the value of

p

. [3]

The equation

f(x) = k

has exactly one solution. Justify why exactly one value of

k

produces this result and state that value of

k

. [1]

Solutions

2QuestionSolving Using the Quadratic FormulaAssessment Practice

6 marks~9 minCriterion C

A quadratic function is given by

f(x) = 2x^2 + 5x + c

, where

c

is a constant.

Given that

f\!\left(\dfrac{1}{2}\right) = 0

, show that

c = -3

. [1]

Find the other solution of

f(x) = 0

. Give your answer as an exact value. [2]

A engineer states that the equation

f(x) = k

has no real solutions for

k < -6

. Justify whether the engineer's claim is correct. [3]

Solutions

3QuestionSolving Using the Quadratic FormulaAssessment Practice

7 marks~11 minCriterion B

A pattern of growing L-shaped figures is made from unit squares. The first three figures are shown in the diagram.

Figure number ( $n$ )	1	2	3
Number of squares ( $S$ )	3	7	13

Write down the number of unit squares in Figure 4. [1]

Find a rule for the total number of unit squares

S

in Figure

n

. Give your rule in the form

S = an^2 + bn + c

, where

a

b

and

c

are integers. Show your method clearly. [3]

A student claims that every figure in the pattern contains an odd number of unit squares. Justify whether this claim is correct for all positive integer values of

n

. [3]

Solutions

4QuestionNature of Roots Using DiscriminantAssessment Practice

10 marks~15 minCriterion D

A basketball player shoots a free throw. The height

h

(in metres) of the ball

t

seconds after release is modelled by

h(t) = -5t^2 + 4t + 2.1.

The hoop is at a height of

3.05

m. The table below shows the height predicted by the model and the actual measured height of the ball at three moments during a real shot.

Time

t

(s):

0.2

0.4

0.6

Predicted

h

(m):

2.70

2.90

2.70

Actual

h

(m):

2.15

2.05

1.78

Show that

h(t) = 3.05

can be written as

5t^2 - 4t + 0.95 = 0,

and hence calculate the discriminant of this equation. [2]

The player adjusts their release so that the new model is

h(t) = -5t^2 + bt + 2.1, \quad b > 0.

Find the minimum value of

b

for which the ball can just reach the hoop height of

3.05

m. Give your answer to 3 significant figures. [3]

Percentage error is defined as

\text{percentage error} = \frac{|\text{predicted} - \text{actual}|}{\text{actual}} \times 100\%.

Calculate the percentage error at each of

t = 0.2

t = 0.4

, and

t = 0.6

. Give each answer to 1 decimal place. [3]

Using your percentage errors from part (c), justify whether this quadratic model is suitable for predicting the height of the ball throughout the shot. [2]

Solutions

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