Solve x² − 4x − 12 ≤ 0
Quadratic inequality, worked out line by line the way a teacher would write it.
Answer
| Solution | −2 ≤ x ≤ 6 |
| Interval notation | [−2, 6] |
Step-by-step solution
6 steps-
1 Givenx^{2} - 4 x - 12 \le 0
Solve for x.
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2 Factor\left(x - 6\right) \left(x + 2\right) \le 0
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3 Critical points: where the expression is 0 or undefined
A polynomial or a fraction can only change sign where it equals 0 or where it is undefined. Between two neighbouring critical points the sign stays the same, so one test value decides each whole interval.
\text{zeros: } x = 6,\; x = -2The sign can only change at these points, so test one value in each interval between them.
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4 Sign chart: the sign of each factor in each interval
Each row shows the sign of one factor in each interval, and the bottom row multiplies them: an even number of minus signs gives +, an odd number gives −.
\begin{array}{c|ccccc} & \left(-\infty,\, -2\right) & -2 & \left(-2,\, 6\right) & 6 & \left(6,\, \infty\right) \\ \hline x - 6 & - & - & - & 0 & + \\ \hline x + 2 & - & 0 & + & + & + \\ \hline \text{whole} & + & 0 & - & 0 & + \end{array}Test values: x = -3, x = -1, x = 7.
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5 Keep the intervals where the expression is negative
Keep the intervals whose sign matches the inequality: > 0 means positive, < 0 negative. With ≥ or ≤ the zeros count as well, but points where the expression is undefined never do.
-2 \le x \le 6The zeros count too (≤ / ≥), but values where it is undefined never do.
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6 Solution-2 \le x \le 6\qquad \left[ -2, 6 \right]
Check with test values
| x | Left side | Right side | Holds? |
|---|---|---|---|
| −1 | −7 | 0 | ✓ true |
| −3 | 9 | 0 | ✗ false |
Change a number or try your own problem: the solver works it out the same way, with a graph and hint mode.