Question 115403
{{{x=1-7x^2}}} Start with the given equation



{{{7x^2+x-1=0}}} Move all of the terms to the left side



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+x-1=0}}} ( notice {{{a=7}}}, {{{b=1}}}, and {{{c=-1}}})





{{{x = (-1 +- sqrt( (1)^2-4*7*-1 ))/(2*7)}}} Plug in a=7, b=1, and c=-1




{{{x = (-1 +- sqrt( 1-4*7*-1 ))/(2*7)}}} Square 1 to get 1  




{{{x = (-1 +- sqrt( 1+28 ))/(2*7)}}} Multiply {{{-4*-1*7}}} to get {{{28}}}




{{{x = (-1 +- sqrt( 29 ))/(2*7)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (-1 +- sqrt(29))/(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 = (-1 +- sqrt(29))/14}}} Multiply 2 and 7 to get 14


So now the expression breaks down into two parts


{{{x = (-1 + sqrt(29))/14}}} or {{{x = (-1 - sqrt(29))/14}}}



Now break up the fraction



{{{x=-1/14+sqrt(29)/14}}} or {{{x=-1/14-sqrt(29)/14}}}



Simplify



{{{x=-1 / 14+sqrt(29)/14}}} or {{{x=-1 / 14-sqrt(29)/14}}}



So these expressions approximate to


{{{x=0.313226057652465}}} or {{{x=-0.456083200509607}}}



So our solutions are:

{{{x=0.313226057652465}}} or {{{x=-0.456083200509607}}}


Notice when we graph {{{7*x^2+x-1}}}, we get:


{{{ graph( 500, 500, -10.4560832005096, 10.3132260576525, -10.4560832005096, 10.3132260576525,7*x^2+1*x+-1) }}}


when we use the root finder feature on a calculator, we find that {{{x=0.313226057652465}}} and {{{x=-0.456083200509607}}}.So this verifies our answer