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A Quadratic within a Quadratic

Example

A quadratic within a quadratic

Solve (x² + 2x)² - 11(x² + 2x) + 24 = 0.

Solution

Note that x² + 2x and (x² + 2x)²appear in the equation. Let a = x² + 2x and substitute.

				a² - 11a + 24		= 0
				(a - 8)(a - 3)		= 0	Factor.
a - 8		= 0	or	a - 3		= 0
a		= 8	or	a		= 3
x² + 2x		= 8	or	x² + 2x		= 3	Replace a by x² + 2x.
x² + 2x - 8		= 0	or	x² + 2x - 3		= 0
(x - 2)(x + 4)		= 0	or	(x + 3)(x - 1)		= 0
x - 2 = 0	or	x + 4	= 0	or	x + 3	= 0	or	x - 1	= 0
x = 2	or	x	= -4	or	x	= -3	or	x	= 1

Check. The solution set is {-4, -3, 1, 2}.