Clarify your complex basic!

Algebra Level 2

x 2 + 18 x + 90 = 0 \large x^2 + 18x + 90 = 0

Find the value(s) of x x satisfying the equation above.

If the answers are of the form − a i − b -ai - b and a i − b ai - b , with positive a a , submit your answer as a + b a+b .

Clarification : i = − 1 i=\sqrt{-1} .

Bonus : Try to solve it without using the quadratic formula .


The answer is 12.

Ashish Menon by Ashish Menon , 20, India

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2 solutions

Hung Woei Neoh
May 5, 2016

Use the quadratic formula to solve this equation

x 2 + 18 x + 90 = 0 x = − 18 ± 1 8 2 − 4 ( 1 ) ( 90 ) 2 ( 1 ) = − 18 ± 324 − 360 2 = − 18 ± − 36 2 = − 9 ± − 9 = − 9 ± 3 i x^2+18x+90=0\\ x=\dfrac{-18 \pm \sqrt{18^2 - 4(1)(90)}}{2(1)}\\ =\dfrac{-18 \pm \sqrt{324-360}}{2}\\ =\dfrac{-18 \pm \sqrt{-36}}{2}\\ =-9 \pm \sqrt{-9}\\ =-9 \pm 3i

Therefore, x = 3 i − 9 , x = − 3 i − 9 x=3i - 9, \; x=-3i-9

a = 3 , b = 9 a=3,\; b=9 and a + b = 3 + 9 = 12 a+b = 3+9 = \boxed{12}

Bonus:

x 2 + 18 x + 90 = 0 x 2 + 18 x + 81 = − 9 ( x + 9 ) 2 = − 9 x + 9 = ± − 9 x + 9 = ± 3 i x = ± 3 i − 9 x^2+18x+90=0\\ x^2+18x+81=-9\\ (x+9)^2 = -9\\ x+9 = \pm \sqrt{-9}\\ x+9 = \pm 3i\\ x= \pm 3i - 9

Therefore, x = 3 i − 9 , x = − 3 i − 9 x=3i - 9, \; x=-3i-9

a = 3 , b = 9 a=3,\; b=9 and a + b = 3 + 9 = 12 a+b = 3+9 = \boxed{12}

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Yes that is another way :)

Ashish Menon - 5 years, 1 month ago

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You told to try without that formula ..so I did it the other way

Abhiram Rao - 5 years, 1 month ago
Ashish Menon
May 5, 2016

x 2 + 18 x + 90 = 0 x 2 + 18 x + 81 + 9 = 0 ( x + 9 ) 2 − 9 i 2 = 0 ( x + 9 ) 2 − ( 3 i ) 2 = 0 ( x + 9 + 3 i ) ( x + 9 − 3 i ) = 0 x = ( − 3 i − 9 ) ( or ) ( 3 i − 9 ) ∴ a + b = 3 + 9 = 12 \begin{aligned} x^2 + 18x + 90 & = 0\\ x^2 + 18x + 81 + 9 & = 0\\ {\left(x + 9\right)}^2 - 9i^2 & = 0\\ {\left(x + 9\right)}^2 - {\left(3i\right)}^2 & = 0\\ \left(x + 9 + 3i\right)\left(x + 9 - 3i\right) & = 0\\ x & = \left(-3i - 9\right) (\text{or}) \left(3i - 9\right)\\ \therefore a + b & = 3 + 9\\ & = \boxed{12} \end{aligned}

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