Question 91107
{{{5x^2 + 14x = -3}}}

{{{5x^2 + 14x +3=0}}} Add 3 to 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 {{{5*x^2+14*x+3=0}}} ( notice {{{a=5}}}, {{{b=14}}}, and {{{c=3}}})


{{{x = (-14 +- sqrt( (14)^2-4*5*3 ))/(2*5)}}} Plug in a=5, b=14, and c=3




{{{x = (-14 +- sqrt( 196-4*5*3 ))/(2*5)}}} Square 14 to get 196  




{{{x = (-14 +- sqrt( 196+-60 ))/(2*5)}}} Multiply {{{-4*3*5}}} to get {{{-60}}}




{{{x = (-14 +- sqrt( 136 ))/(2*5)}}} Combine like terms in the radicand (everything under the square root)




{{{x = (-14 +- 2*sqrt(34))/(2*5)}}} 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 = (-14 +- 2*sqrt(34))/10}}} Multiply 2 and 5 to get 10


So now the expression breaks down into two parts


{{{x = (-14 + 2*sqrt(34))/10}}} or {{{x = (-14 - 2*sqrt(34))/10}}}



Now break up the fraction



{{{x=-14/10+2*sqrt(34)/10}}} or {{{x=-14/10-2*sqrt(34)/10}}}



Simplify



{{{x=-7 / 5+sqrt(34)/5}}} or {{{x=-7 / 5-sqrt(34)/5}}}



So these expressions approximate to


{{{x=-0.23380962103094}}} or {{{x=-2.56619037896906}}}



So our solutions are:

{{{x=-0.23380962103094}}} or {{{x=-2.56619037896906}}}


Notice when we graph {{{5*x^2+14*x+3}}}, we get:


{{{ graph( 500, 500, -12.5661903789691, 9.76619037896906, -12.5661903789691, 9.76619037896906,5*x^2+14*x+3) }}}


when we use the root finder feature on a calculator, we find that {{{x=-0.23380962103094}}} and {{{x=-2.56619037896906}}}.So this verifies our answer