Question 101192


{{{4x^2+2x-3=0}}} Start with the given equation



{{{4x^2+2x=3}}} Add 3 to both sides
{{{4(x^2+0.5x)=3}}} Factor out the leading coefficient 4.  This step is important since we want the {{{x^2}}} coefficient to be equal to 1.




Take half of the x coefficient 0.5 to get 0.25 (ie {{{0.5/2=0.25}}})

Now square 0.25 to get 0.0625 (ie {{{(0.25)^2=0.0625}}})




{{{4(x^2+0.5x+0.0625)=3+0.0625(4)}}} Add this result (0.0625) to the expression {{{x^2+0.5x}}}  inside the parenthesis. Now the expression {{{x^2+0.5x+0.0625}}}  is a perfect square trinomial. Now add the result (0.0625)(4) (remember we factored out a 4) to the right side.




{{{4(x+0.25)^2=3+0.0625(4)}}} Factor {{{x^2+0.5x+0.0625}}} into {{{(x+0.25)^2}}} 



{{{4(x+0.25)^2=3+0.25}}} Multiply 0.0625 and 4 to get 0.25




{{{4(x+0.25)^2=3.25}}} Combine like terms on the right side


{{{(x+0.25)^2=0.8125}}} Divide both sides by 4



{{{x+0.25=0+-sqrt(0.8125)}}} Take the square root of both sides


{{{x=-0.25+-sqrt(0.8125)}}} Subtract 0.25 from both sides to isolate x.


So the expression breaks down to

{{{x=-0.25+sqrt(0.8125)}}} or {{{x=-0.25-sqrt(0.8125)}}}



So our answer is approximately

{{{x=0.651387818865997}}} or {{{x=-1.151387818866}}}


Here is visual proof


{{{ graph( 500, 500, -10, 10, -10, 10, 4x^2+2x-3) }}} graph of {{{y=4x^2+2x-3}}}



When we use the root finder feature on a calculator, we would find that the x-intercepts are {{{x=0.651387818865997}}} and {{{x=-1.151387818866}}}, so this verifies our answer.