Question 134326
{{{x^2+3 x=5}}} Start with the given equation



Take half of the x coefficient {{{3}}} to get {{{3/2}}} (ie {{{(1/2)(3)=3/2}}}).


Now square {{{3/2}}} to get {{{9/4}}} (ie {{{(3/2)^2=(3/2)(3/2)=9/4}}})





{{{x^2+3x+9/4-9/4=5}}} Now add and subtract this value on the left side. Doing both the addition and subtraction of {{{9/4}}} does not change the equation




{{{(x+3/2)^2-9/4=5}}} Now factor {{{x^2+3x+9/4}}} to get {{{(x+3/2)^2}}}



{{{(x+3/2)^2=9/4+5}}} Add {{{9/4}}} to both sides



{{{(x+3/2)^2=29/4}}} Combine like terms




*[Tex \LARGE x+\frac{3}{2}=\pm \sqrt{\frac{29}{4}}] Take the square root of both sides



*[Tex \LARGE x+\frac{3}{2}=\pm \frac{\sqrt{29}}{2}] Simplify


*[Tex \LARGE x=-\frac{3}{2}\pm \frac{\sqrt{29}}{2}]  Subtract {{{3/2}}} from both sides


*[Tex \LARGE x=\frac{-3\pm\sqrt{29}}{2}]  Combine the fractions




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Answer:



So our solutions are 


*[Tex \LARGE x=\frac{-3+\sqrt{29}}{2}] or *[Tex \LARGE x=\frac{-3-\sqrt{29}}{2}]