Solving a quadratic inequality

Algebra

(1MA1) · Higher

ORIGINAL

Solve the equation, sketch the parabola, then read the region — the sketch is what stops sign errors.

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Question

Solve

x^2 - x - 6 > 0

Every step worked, with the reasoning.

1
$x^2 - x - 6 = 0$
Solve the equation first to find the critical values.
2
$(x - 3)(x + 2) = 0$
Factorise.
3
$x = 3$ or $x = -2$
The critical values — where the curve crosses the $x$ -axis.
4
Sketch: a $\cup$ -shaped parabola with roots at $-2$ and $3$ ; we want where it is $> 0$ (above the axis).
A positive $x^2$ coefficient means the curve is above the axis OUTSIDE the roots.
5
$x < -2$ or $x > 3$
Read off the two regions where the curve is above the $x$ -axis.

Answer: $x < -2$ or $x > 3$