y = 4x^2 - 2x - 5 = 0
Step One
Put brackets around the first 2 terms. Put the five outside the brackets.
(4x^2 - 2x) - 5 = 0
Step Two
Treat 4 as a common factor put it on the left bracket.
4(x^2 - 2/4 x) - 5 = 0
4(x^2 - 1/2 x) - 5 = 0
Step Three
Take 1/2 the linear term and square it. -1/2 x is the linear term. Drop the x for this step. Leave 1/2 the linear term squared added inside the brackets.
4(x^2 - 1/2x + (1/2*1/2)^2 ) - 5 = 0
Step Four
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You've got an unbalanced equation. It will not break down to what you started with. You have to adjust it with a number after the five.
4(x^2 - 1/2 x + (1/4)^2 ) - 5 - 4 * (1/4)^2
4(x^2 - 1/2 x + 1/16 ) - 5 - 1/4
4(x^2 - 1/2 x + 1/16 ) - 5 - 1/4
4(x^2 - 1/2 x + 1/16 ) - 21/4
Step five
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Express what is inside the brackets as a perfect square.
4(x - 1/4)^2 - 21/4 = 0
This is the vertex form of the equation. Now to get the roots which is what you are after.
Step six
Transfer 21/4 to the right hand side.
4(x - 1/4)^2 = 21/4
Step seven
Divide by 4
(x - 1/4)^2 = 21/16
Step eight
Take the square root of both sides.
x - 1/4 = +/- sqrt(21)/4 Transfer the 1/4 over to the right.
x = [1 +/- sqrt(21) )/ 4
That gives you two answers
x = [1 + sqrt(21) ) / 4 <<<<< sqrt answer
x = 1.3945 <<<<< decimal answer.
or
x = [1 - sqrt(21)] / 4 <<<< sqrt answer
x = - 0.8956 Decimal answer
Comment
If you use the quadratic formula, you get
a = 4
b = - 2
c = - 5
x = [- b +/- sqrt ( b^2 - 4ac) ] / 2a
x = [ 2 +/- sqrt ( (-2)^2 - 4((4)(-5))]/2*4
x = [ 2 +/- sqrt ( 4 + 80 ) ] / 8
x = [2 +/- sqrt (84) ] / 8
x = [2 +/- sqrt (4*21) ] /8
x = [2 +/- 2*sqrt (21) ] / 8
x = 2 [1 +/- sqrt(21)] / 8
x = [1 +/- sqrt(21) ] /4 Now you can look above to decide on your answer
