Question 88512
{{{z^2-8z=-5}}}


{{{z^2-8z+5=0}}} Add 5 to both sides


Now let's use the quadratic formula to solve for z:



Starting with the general quadratic


{{{az^2+bz+c=0}}}


the general solution using the quadratic equation is:


{{{z = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}


So lets solve {{{z^2-8*z+5=0}}} ( notice {{{a=1}}}, {{{b=-8}}}, and {{{c=5}}})


{{{z = (--8 +- sqrt( (-8)^2-4*1*5 ))/(2*1)}}} Plug in a=1, b=-8, and c=5




{{{z = (8 +- sqrt( (-8)^2-4*1*5 ))/(2*1)}}} Negate -8 to get 8




{{{z = (8 +- sqrt( 64-4*1*5 ))/(2*1)}}} Square -8 to get 64




{{{z = (8 +- sqrt( 64+-20 ))/(2*1)}}} Multiply {{{-4*5*1}}} to get {{{-20}}}




{{{z = (8 +- sqrt( 44 ))/(2*1)}}} Combine like terms in the radicand (everything under the square root)




{{{z = (8 +- 2*sqrt(11))/(2*1)}}} Simplify the square root




{{{z = (8 +- 2*sqrt(11))/2}}} Multiply 2 and 1 to get 2


So now the expression breaks down into two parts


{{{z = (8 + 2*sqrt(11))/2}}} or {{{z = (8 - 2*sqrt(11))/2}}}



Which approximate to


{{{z=7.3166247903554}}} or {{{z=0.6833752096446}}}



So our solutions are:

{{{z=7.3166247903554}}} or {{{z=0.6833752096446}}}


Notice when we graph {{{x^2-8*x+5}}} (just replace z with x), we get:


{{{ graph( 500, 500, -9.3166247903554, 17.3166247903554, -9.3166247903554, 17.3166247903554,1*x^2+-8*x+5) }}}


when we use the root finder feature on a calculator, we find that {{{x=7.3166247903554}}} and {{{x=0.6833752096446}}}.So this verifies our answer