If $i^2=-1$, then $4+i$, and $4-i$ are roots of which of the following equations?
- $x^2+8x-17=0$
- $x^2-8x+17=0$
- $x^2-8x-17=0$
- $x^2+10x-8=0$
- $x^2-8x+8=0$
Reveal answer
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B
Correct answer
Explanation
If roots are 4+i and 4-i, the equation is (x - (4+i))(x - (4-i)) = 0. This is ((x-4) - i)((x-4) + i) = 0, which is (x-4)^2 - i^2 = 0. Since i^2 = -1, (x^2 - 8x + 16) - (-1) = 0, so x^2 - 8x + 17 = 0.
AI explanation
Using the sum and product of roots formulas, we add the roots to get (4+i) + (4-i) = 8, which means the coefficient of the x term is -8. We multiply the roots to find the constant term, where (4+i)(4-i) equals 4^2 minus i^2, resulting in 16 - (-1) = 17. Substituting these calculated values into the standard quadratic form x^2 - (sum)x + (product) = 0 gives the equation x^2 - 8x + 17 = 0.