Question 132297
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 {{{3*x^2-5*x-7=0}}} ( notice {{{a=3}}}, {{{b=-5}}}, and {{{c=-7}}})





{{{x = (--5 +- sqrt( (-5)^2-4*3*-7 ))/(2*3)}}} Plug in a=3, b=-5, and c=-7




{{{x = (5 +- sqrt( (-5)^2-4*3*-7 ))/(2*3)}}} Negate -5 to get 5




{{{x = (5 +- sqrt( 25-4*3*-7 ))/(2*3)}}} Square -5 to get 25  (note: remember when you square -5, you must square the negative as well. This is because {{{(-5)^2=-5*-5=25}}}.)




{{{x = (5 +- sqrt( 25+84 ))/(2*3)}}} Multiply {{{-4*-7*3}}} to get {{{84}}}




{{{x = (5 +- sqrt( 109 ))/(2*3)}}} Combine like terms in the radicand (everything under the square root)






{{{x = (5 +- sqrt(109))/6}}} Multiply 2 and 3 to get 6


Now break up the expression



{{{x = (5 + sqrt(109))/6}}}  or {{{x = (5 - sqrt(109))/6}}} 



So these expressions approximate to


{{{x=2.57338441815176}}} or {{{x=-0.906717751485092}}}



So our solutions are:

{{{x=2.5734}}} or {{{x=-0.9067}}}



So that means that the x-intercepts are 


(-0.9067,0) and (2.5734,0)






Notice when we graph {{{3*x^2-5*x-7}}}, we get:


{{{ graph( 500, 500, -10.9067177514851, 12.5733844181518, -10.9067177514851, 12.5733844181518,3*x^2+-5*x+-7) }}}


when we use the root finder feature on a calculator, we find that {{{x=2.5734}}} and {{{x=-0.9067}}}.So this verifies our answer