# Chapter 6 - Section 6.4 - Logarithmic Functions - 6.4 Assess Your Understanding: 106

$x_1=-\dfrac{1}{2} - \dfrac{\sqrt{85}}{2} \\x_2=-\dfrac{1}{2} + \dfrac{\sqrt{85}}{2}$

#### Work Step by Step

RECALL: $\log_a{x}=y \longrightarrow a^y=x, a\gt0, a\ne1$ Use the rule above to obtain: $\log_5{(x^2+x+4)}=2 \\\longrightarrow 5^2=x^2+x+4 \\25=x^2+x+4$ Subtract by 25 on both sides of the equation to obtain: $25-25=x^2+x+4-25 \\0=x^2+x-21$ Use the quadratic formula, $x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$, where $a = 1, b=1,$ and $c=-21$ to obtain: $x=\dfrac{-1 \pm \sqrt{1^2-4(1)(-21)}}{2(1)} \\x=\dfrac{-1 \pm \sqrt{1-(-84)}}{2} \\x=\dfrac{-1 \pm \sqrt{1+84}}{2} \\x=\dfrac{-1 \pm \sqrt{85}}{2}$ Split the solutions to obtain: $x_1=-\dfrac{1}{2} - \dfrac{\sqrt{85}}{2} \\x_2=-\dfrac{1}{2} + \dfrac{\sqrt{85}}{2}$

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