Question

Math

Posted 4 months ago

`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 $m+n$.`

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Answer from Sia

Posted 4 months ago

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 $m + n$.

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