Solve an equation that reduces to quadratic form practice questions | Mathematical Methods (Year 11, VCE Units 1 & 2, Quadratic functions, Quadratic equations, inequalities and intersections)

(a)1 mark

Which substitution replaces the common even-power block by \(u\), so the highest power becomes \(u^2\) directly, without an additional scale or sign change?

u=x

u=2x^2

u=x^2

u=x^4

(b)2 marks

Solve the resulting quadratic equation in \(u\). Enter the two values in ascending order

\(u_1\) \(u_2\)

(c)2 marks

Use the definition of \(u\) to find all real solutions of the original equation. Enter them in ascending order

\(x_1\) \(x_2\)

(d)1 mark

Which statement gives the exact validity check for the candidate values?

Reject \(u=-9\)

This value is negative, but \(u=x^2\) is non-negative

Reject \(u=4\)

The original equation is not satisfied after back-substitution

Both candidate values of \(u\) give real solutions for \(x\) in the original equation