Hint
For a real \(\displaystyle x, x^{2}\ge 0\). Solve the quadratic in \(\displaystyle x^{2}\) first.
Worked solution
(a)
\(\left(x^{2} - 49\right) \left(x^{2} - 4\right) = 0\) gives \(\displaystyle x^{2}=a^{2}\) or \(\displaystyle b^{2}\); each is positive and gives two roots.
Answer: \(\left\{-7, -2, 2, 7\right\}\)
(b)
Take the positive square root of the larger u-root.
\(7\)
Answer: \(7\)
(c)
Pair each positive root with its negative.
\(196\)
Answer: \(196\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Solve the quadratic in \(\displaystyle x^{2}\) and then take both signs for each positive value.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The substituted roots are not automatically x-roots.
(b) · 2 marks
- 1 mark: Take the positive square root of the larger u-root.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not discard the negative x branch of a positive square.
(c) · 2 marks
- 1 mark: Pair each positive root with its negative.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: The substituted roots are not automatically x-roots.
Total: 6 marks.