Solving Quadratic Equations

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Solving Quadratic Equations — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionNature of Roots Using DiscriminantAssessment Practice

5 marks~8 minCriterion A

A quadratic function is given by

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

, where

b

and

c

are integers.

The graph of

f(x)

passes through the point

(1, 0)

and has its axis of symmetry at

x = 3

Write down the two roots of

f(x) = 0

. [1]

Find the values of

b

and

c

. [2]

The equation

f(x) = k

has no real solutions. Justify whether a value of

k = -5

would satisfy this condition, using the vertex of the parabola. [2]

Solutions

2QuestionSolving Using the Quadratic FormulaAssessment Practice

6 marks~9 minCriterion B

The diagram shows Figures 1 to 4 of a pattern made from small squares arranged in an L-shape.

Figure 1: 3 squares, Figure 2: 7 squares, Figure 3: 13 squares, Figure 4: 21 squares.

Write down the number of small squares in Figure 5. [1]

Find a rule for the number of small squares

S

in Figure

n

, giving your answer in the form

S = n^2 + bn + c

, where

b

and

c

are integers to be found. [3]

A student claims that no figure in this pattern can contain exactly 100 small squares. Justify whether the student is correct. [2]

Solutions

3QuestionSolving Using the Quadratic FormulaAssessment Practice

6 marks~9 minCriterion D

A basketball player shoots a free throw. The height

h

metres of the ball above the ground,

t

seconds after release, is modelled by

h(t) = -4.9t^2 + 8t + 2, \quad t \geq 0.

A motion sensor records the following observed heights.

t

(seconds):

0.4

\quad

0.8

\quad

1.2

Observed

h

(m):

4.1

\quad

5.0

\quad

3.6

Calculate the predicted heights given by the model at

t = 0.4

t = 0.8

, and

t = 1.2

. Give each value to 2 decimal places. [1]

The axis of symmetry of the parabola is

t = \dfrac{8}{9.8}

. Calculate the maximum height of the ball. Give your answer in metres to 3 significant figures. [2]

The player claims the ball reaches a maximum height of more than 5 metres. Assess whether the model supports this claim, and justify your reasoning using your answers to parts (a) and (b). [3]

Solutions

4QuestionNature of Roots Using DiscriminantAssessment Practice

10 marks~15 minCriterion C

Consider the quadratic function

f(x) = x^2 + kx + 9

, where

k

is a real constant.

Find the set of values of

k

for which

f(x) = 0

has two distinct real roots, giving your answer in set notation. [3]

For the specific case

k = 8

, show that the vertex of the graph of

y = f(x)

lies below the

x

-axis, and hence state the number of

x

-intercepts of the graph. [3]

A student claims: "Because the coefficient of

x^2

is positive and

f(0) = 9 > 0

, the graph of

y = x^2 + kx + 9

can never cross the

x

-axis for any real value of

k

."

Critique this claim. Use the discriminant to identify a specific value of

k

for which the graph crosses the

x

-axis at two distinct points, and explain why the student's reasoning is flawed. [4]

Solutions

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