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# help algebra

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The roots of the quadratic equation z^2 + az + b = 0 are -7 + 3i and -7 - 3i.  What is a + b?

Jul 2, 2021

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The quadratic formula is, $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$. First let's focus on the -7. To make -b/2 = -7, b would have to be b = 14 to satisfy. So already have,

$$z^2 + 14z + b = 0$$

Now that we know b, we need to have $$\frac{\sqrt{196-4c}}{2} = 3i$$. Square both sides and further solve,

$$\frac{196 - 4c}{4} = -9$$

$$196 - 4c = -36$$

$$4c = 232$$

$$c = 58$$

So completing the equation we have, $$z^2 + 14z + 58 = 0$$, and we can check using the quadratic formula to see that the roots are $$-7 \pm 3i$$. So,$$a + b = 14 + 58 =$$ 72

Jul 3, 2021