If $x^{2}+b x+b+3=0$ has roots of the form $\frac{-b \pm \sqrt{5}}{2}$, where $b0$, then $b=m+\sqrt{n}$ for positive integers $m, n$. Find $m+n$.

Question

If $x^{2}+b x+b+3=0$ has roots of the form $\frac{-b \pm \sqrt{5}}{2}$, where $b>0$, then $b=m+\sqrt{n}$ for positive integers $m, n$. Find $m+n$.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Given the quadratic equation $ x^{2} + bx + b + 3 = 0 $ and its roots $ \frac{-b \pm \sqrt{5}}{2} $, we can use the quadratic formula to find the value of $ b $. ‖ ‖ step 2 ⋮ The quadratic formula for the roots of $ ax^2 + bx + c = 0 $ is $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $. Comparing this with the given roots, we have $ \sqrt{b^2 - 4(b + 3)} = \sqrt{5} $. ‖ ‖ step 3 ⋮ Squaring both sides of the equation $ \sqrt{b^2 - 4(b + 3)} = \sqrt{5} $ gives $ b^2 - 4b - 12 = 5 $. ‖ ‖ step 4 ⋮ Solving $ b^2 - 4b - 12 = 5 $ for $ b $ gives $ b^2 - 4b - 17 = 0 $. ‖ ‖ step 5 ⋮ The solutions for $ b $ are $ b = 2 - \sqrt{21} $ and $ b = 2 + \sqrt{21} $. Since $ b > 0 $, we take the positive solution $ b = 2 + \sqrt{21} $. ‖ ‖ step 6 ⋮ We have $ b = m + \sqrt{n} $ where $ m = 2 $ and $ n = 21 $. ‖ ‖ step 7 ⋮ To find $ m + n $, we add $ m $ and $ n $ to get $ 2 + 21 $. ‖ ∻Answer∻ ⚹ $ m + n = 23 $ ⚹ ∻Key Concept∻ ⚹Quadratic Equations and Roots⚹ ∻Explanation∻ ⚹The roots of a quadratic equation can be found using the quadratic formula. By comparing the given roots with the formula, we can solve for the unknown coefficient $ b $. Once $ b $ is found, we can express it in the form $ b = m + \sqrt{n} $ and find the sum $ m + n $.⚹◊What is the relationship between the roots of a quadratic equation and the coefficients in the equation?⍭ Generate me a similar question◊

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