Mathematical Methods — Question 15

QCAA 2025, Paper 1 · 6 marks

Q15 · 2025 · Technology-freeSimple familiar6 marks

QUESTION 15 (6 marks)

Determine all possible solutions for the equation

(\log_2(x))^2=\log_2(x^2)+8

. Evaluate the reasonableness of one solution.

WORKED SOLUTION

6 marks

ANSWER

x=16

x=\frac14

; for example,

x=16

satisfies the original equation.

Worked solution

(\log_2x)^2-\log_2(x^2)-8=0.

Using

\log_2(x^2)=2\log_2x

(\log_2x)^2-2\log_2x-8=0.

Let

y=\log_2x

. Then

y^2-2y-8=0=(y-4)(y+2),

y=4

y=-2

. Hence

x=2^4=16\quad\text{or}\quad x=2^{-2}=\frac14.

To evaluate

x=16

in the original equation:

\text{LHS}=(\log_2 16)^2=4^2=16,

\text{RHS}=\log_2(16^2)+8=\log_2(2^8)+8=16.

Therefore

x=16

is a reasonable solution.

correctly applies an appropriate logarithmic law to convert the

x^2

term to a term in

x

[1 mark]

realises the need to solve a quadratic equation

[1 mark]

determines a solution

[1 mark]

determines a second solution

[1 mark]

substitutes the chosen solution into the equation

[1 mark]

evaluates the reasonableness of one possible solution

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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