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Each solution to x^2 + 5x + 18 = 0 can be written in the form a + bi where a and b are real numbers. What is a^2 + b^2?

 Feb 16, 2022
 #1
avatar+854 
+2

Instead of doing the entire problem by getting each root exactly, why not try and use Vieta's formula to get ab and a + b instead? 

ab = 18 and a + b = -5. Thus, a^2 + b^2 = (a + b)^2 - 2ab. Then a^2 + b^2 = 25 - 36.

 

Thus, a^2 + b^2 = -11. smiley

 Feb 16, 2022
 #4
avatar+854 
+8

Oh yes! Thank you for your input everyone! But I caught a quick error! The solution ITSELF is a + bi, the solutions AREN'T a, and b. Thus, we might have to use the quadratic formula to find the solutions.

 

Applying the quadratic formula \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\). We get -5 +- sqrt(25 - 72) / 2. 

Thus the two solutions are \(-{5\over2}+{\sqrt{-47}\over2}\) and \(-{5\over2} - {\sqrt{-47}\over2}\)

 

In this case, it doesn't matter whether b is negative or positive since we are squaring it eventually. a is 5/2. Thus, a^2 = 25/4 and b^2 = -47 or -188/4. 

 

That means, \(a^2 + b^2 = -{163\over4}\), which approximates to -40.75.

smiley

proyaop  Feb 16, 2022
 #2
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Using Vieta's, a+b= -5 and ab = 18. a^2 + b^2 = (a+b)^2 - 2ab.                Reasoning: (a+b)^2 = a^2 + 2ab + b^2, so subtracting 2ab will get a^2 + b^2, our requested sum. 

 

So our final answer is 25 - 36 = -11.

 Feb 16, 2022
 #3
avatar+23200 
+2

If you don't know Vieta's formula:

1)  Use the quadratic formula to find the two solutions.

2)  Square each solution and add them together.

 

It won't be as quick, but you won't need to memorize another formula.

 Feb 16, 2022

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