Question 134476

{{{y=x^2+x-5}}} Start with the given equation



{{{y+5=x^2+x}}} Add {{{5}}} to both sides




Take half of the x coefficient {{{1}}} to get {{{1/2}}} (ie {{{(1/2)(1)=1/2}}}).


Now square {{{1/2}}} to get {{{1/4}}} (ie {{{(1/2)^2=(1/2)(1/2)=1/4}}})





{{{y+5=x^2+x+1/4-1/4}}} Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of {{{1/4}}} does not change the equation




{{{y+5=(x+1/2)^2-1/4}}} Now factor {{{x^2+1x+1/4}}} to get {{{(x+1/2)^2}}}



{{{y+5=(x+1/2)^2-1/4}}} Multiply



{{{y=(x+1/2)^2-1/4-5}}} Now add {{{+5}}} to both sides to isolate y



{{{y=(x+1/2)^2-21/4}}} Combine like terms




{{{(x+1/2)^2-21/4=0}}} Now to solve for x, let {{{y=0}}}



{{{(x+1/2)^2=+21/4}}} Add {{{21/4}}} to both sides



*[Tex \LARGE x+\frac{1}{2}=\pm \sqrt{\frac{21}{4}}] Take the square root of both sides



*[Tex \LARGE x+\frac{1}{2}=\pm \frac{\sqrt{21}}{2}] Simplify



*[Tex \LARGE x=-\frac{1}{2}\pm \frac{\sqrt{21}}{2}]  Subtract {{{1/2}}} from both sides



*[Tex \LARGE x=\frac{-1\pm\sqrt{21}}{2}]  Combine the fractions



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Answer:


So our solutions are *[Tex \LARGE x=\frac{-1+\sqrt{21}}{2}] or *[Tex \LARGE x=\frac{-1-\sqrt{21}}{2}]