Question 113940


{{{4x^2-20x=-25}}} Start with the given equation



{{{4(x^2-5x)=-25}}} 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 -5 to get -2.5 (ie {{{-5/2=-2.5}}})

Now square -2.5 to get 6.25 (ie {{{(-2.5)^2=6.25}}})




{{{4(x^2-5x+6.25)=-25+6.25(4)}}} Add this result (6.25) to the expression {{{x^2-5x}}}  inside the parenthesis. Now the expression {{{x^2-5x+6.25}}}  is a perfect square trinomial. Now add the result (6.25)(4) (remember we factored out a 4) to the right side.




{{{4(x-2.5)^2=-25+6.25(4)}}} Factor {{{x^2-5x+6.25}}} into {{{(x-2.5)^2}}} 



{{{4(x-2.5)^2=-25+25}}} Multiply 6.25 and 4 to get 25




{{{4(x-2.5)^2=0}}} Combine like terms on the right side


{{{(x-2.5)^2=0}}} Divide both sides by 4



{{{x-2.5=0+-sqrt(0)}}} Take the square root of both sides


{{{x-2.5=0}}} Take the square root of zero to get zero


{{{x=2.5}}} Add 2.5 to both sides to isolate x.



So our answer is

{{{x=2.5}}} 



Here is visual proof


{{{ graph( 500, 500, -10, 10, -10, 10, 4x^2-20x+25) }}} graph of {{{y=4x^2-20x+25}}}


Here we can see that the x-intercept is {{{x=2.5}}}, so this verifies our answer.