First slide

Theory of equations

Question

For the equation $|x^{2}| + |x| - 6 = 0$ , the roots are

Moderate

A

One and only one real number
B

Real with sum one
C

Real with sum zero
D

Real with product zero

Solution

When $x < 0, |x| = - x$
$∴ Equation is x^{2} - x - 6 = 0 \Rightarrow x = - 2, 3$
$∵ x < 0, ∴ x = - 2$ is the solution.
When $x \geq 0, |x| = x$
$∴ Equation is x^{2} + x - 6 = 0 \Rightarrow x = 2, - 3$
$∵ x \geq 0, ∴ x = 2$ is the solution.
Hence $x = 2, - 2$ are the solutions and their sum is zero.

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