Question 113973
{{{7x^2-32x=31}}} Start with the given equation



{{{7x^2-32x-31=0}}} Subtract 31 from both sides


Let's use the quadratic formula to solve for x:



Starting with the general quadratic


{{{ax^2+bx+c=0}}}


the general solution using the quadratic equation is:


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




So lets solve {{{7*x^2-32*x-31=0}}} ( notice {{{a=7}}}, {{{b=-32}}}, and {{{c=-31}}})





{{{x = (--32 +- sqrt( (-32)^2-4*7*-31 ))/(2*7)}}} Plug in a=7, b=-32, and c=-31




{{{x = (32 +- sqrt( (-32)^2-4*7*-31 ))/(2*7)}}} Negate -32 to get 32




{{{x = (32 +- sqrt( 1024-4*7*-31 ))/(2*7)}}} Square -32 to get 1024  (note: remember when you square -32, you must square the negative as well. This is because {{{(-32)^2=-32*-32=1024}}}.)




{{{x = (32 +- sqrt( 1024+868 ))/(2*7)}}} Multiply {{{-4*-31*7}}} to get {{{868}}}




{{{x = (32 +- sqrt( 1892 ))/(2*7)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (32 +- 2*sqrt(473))/(2*7)}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)




{{{x = (32 +- 2*sqrt(473))/14}}} Multiply 2 and 7 to get 14


So now the expression breaks down into two parts


{{{x = (32 + 2*sqrt(473))/14}}} or {{{x = (32 - 2*sqrt(473))/14}}}



Now break up the fraction



{{{x=+32/14+2*sqrt(473)/14}}} or {{{x=+32/14-2*sqrt(473)/14}}}



Simplify



{{{x=16 / 7+sqrt(473)/7}}} or {{{x=16 / 7-sqrt(473)/7}}}



So these expressions approximate to


{{{x=5.39265188156165}}} or {{{x=-0.821223310133078}}}



So our solutions are:

{{{x=5.39265188156165}}} or {{{x=-0.821223310133078}}}


Notice when we graph {{{7*x^2-32*x-31}}}, we get:


{{{ graph( 500, 500, -10.8212233101331, 15.3926518815616, -10.8212233101331, 15.3926518815616,7*x^2+-32*x+-31) }}}


when we use the root finder feature on a calculator, we find that {{{x=5.39265188156165}}} and {{{x=-0.821223310133078}}}.So this verifies our answer