Question 133289
{{{1 z^2-5 z-7=0}}} Start with the given expression



{{{1(z^2-5z-7)=0}}} Factor out the leading coefficient {{{1}}}



Take half of the z coefficient {{{-5}}} to get {{{-5/2}}} (ie {{{(1/2)(-5)=-5/2}}}).


Now square {{{-5/2}}} to get {{{25/4}}} (ie {{{(-5/2)^2=(-5/2)(-5/2)=25/4}}})





{{{1(z^2-5z+25/4-25/4-7)=0}}} Now add and subtract this value inside the parenthesis. Notice how {{{25/4-25/4=0}}}. Since we're adding 0, we're not changing the equation.




{{{1((z-5/2)^2-25/4-7)=0}}} Now factor {{{z^2-5z+25/4}}} to get {{{(z-5/2)^2}}}




{{{1((z-5/2)^2-53/4)=0}}} Combine like terms




{{{1(z-5/2)^2+1(-53/4)=0}}} Distribute




{{{1(z-5/2)^2-53/4=0}}} Multiply





{{{1(z-5/2)^2=53/4}}} Add {{{53/4}}} to both sides





{{{(z-5/2)^2=(53/4)/(1)}}} Divide both sides by {{{1}}}




{{{(z-5/2)^2=53/4}}} Reduce




*[Tex \LARGE z-\frac{5}{2}=\pm \sqrt{\frac{53}{4}}] Take the square root of both sides




*[Tex \LARGE z-\frac{5}{2}=\pm \frac{\sqrt{53}}{2}] Simplify




*[Tex \LARGE z=\frac{5}{2}\pm \frac{\sqrt{53}}{2}] Add {{{5/2}}} to both sides




*[Tex \LARGE z= \frac{5\pm\sqrt{53}}{2}] Combine the fractions




Break up the expression



*[Tex \LARGE z= \frac{5+\sqrt{53}}{2}] or *[Tex \LARGE z= \frac{5-\sqrt{53}}{2}]




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Answer:



So our solutions are 



*[Tex \LARGE z= \frac{5+\sqrt{53}}{2}] or *[Tex \LARGE z= \frac{5-\sqrt{53}}{2}]