Porisma

Worked solution

Read the same interval from a parabola and a sign line

Find the range of values of x such that x2+2x−15<0x^2+2x-15<0.

CJC · Lecture · Example 1

Method 1: graph the polynomial

y=x2+2x−15=(x+5)(x−3)y=x^2+2x-15=(x+5)(x-3). The roots are x=−5 and x=3. Negative polynomial values mean the graph lies below the x-axis, which occurs for −5<x<3.

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Method 2: test between critical values

Factor (x+5)(x−3)<0(x+5)(x-3)<0. Mark its zeros −5,3 and test convenient inputs in (−∞,−5)(-\infty,-5), (−5,3)(−5,3) and (3,∞)(3,\infty); for example −6,0,4 give positive, negative, positive values. Hence −5<x<3. The source test-point paragraph accidentally writes (x−5)(x+3)(x-5)(x+3); test the original (x+5)(x−3)(x+5)(x-3) instead, consistent with its roots and final answer.

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The parabola crosses at −5 and 3 and is negative between them. The source sign line has signs plus, minus, plus and open endpoints at −5 and 3.
The parabola crosses at −5 and 3 and is negative between them. The source sign line has signs plus, minus, plus and open endpoints at −5 and 3.
The parabola crosses at −5 and 3 and is negative between them. The source sign line has signs plus, minus, plus and open endpoints at −5 and 3.

Common mistakes

Strict inequalities exclude zero-valued endpoints. The misprinted test expression has different roots; use the original factors.

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