The roots of the equation $x^2-8x -16= 0$
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are both positive
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are both negative
- one root lies in the interval $(-1, 0)$
- lie in the interval $(-2, 10)$
Reveal answer
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Correct answer
Explanation
For x^2 - 8x - 16 = 0, the roots are x = [8 +/- sqrt(64 - 4(1)(-16))] / 2 = [8 +/- sqrt(128)] / 2 = 4 +/- 4*sqrt(2). Roots are approx 4 + 5.65 = 9.65 and 4 - 5.65 = -1.65. Both roots lie in the interval (-2, 10).
AI explanation
Rewrite the given equation as x^2 - 8x = 16 and complete the square by adding 16 to both sides to get (x - 4)^2 = 32. Taking the square root of both sides gives x - 4 = 4 times the square root of 2, which means x = 4 plus or minus 5.66. This yields the two roots as approximately 9.66 and -1.66, both of which fall safely inside the range from -2 to 10. The roots lie in the interval (-2, 10).