SOLUTION: Can the following equation be solved by the method "completing the square"
4x^2 – 7x – 2 = 0
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Question 252802: Can the following equation be solved by the method "completing the square"
4x^2 – 7x – 2 = 0
Found 2 solutions by ptaylor, Theo:
Answer by ptaylor(2198) (Show Source): You can put this solution on YOUR website!
4x^2 – 7x – 2 = 0 add 2 to each side
4x^2-7x=2 divide each term by 4
x^2-(7/4)x=1/2 take half of the x coefficient and square it and we get 49/64 and add to each side:
x^2-(7/4)x+49/64=32/64+49/64=81/64
(x-7/8)^2=81/64 take square root of each side
x-7/8=+or-9/8
x=7/8+or-9/8
x=-2/8=-1/4
and
x=16/8=2
The answer is "yes"
Hope this helps---ptaylor
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
yes.
all quadratic equations can be solved by completing the squares method. some are easier than others. there are some steps that you have to go through to make the a term in the equation equal to 1.
the standard form of a quadratic equation is ax^2 + bx + c = 0 where:
a = coefficient of the x^2 term.
b = coeficient of the x term.
c = a constant.
here's how your equation is solved using the completing the squares method.
add 2 to both sides of the equation to get:
divide both sides of the equation by 4 to get:
take half of (7/4) to get (7/8)
take (7/8) squared to get (7/8)^2 = (49/64)
your equation becomes:
see bottom for explanation of how this was derived.
take square root of both sides to get:
= +/-
add (7/8) to both sides to get:
+/-
this makes x equal to:
or:
this makes x equal to:
2 or -.25
substitute in original equation to confirm these answers.
original equation is:
when x = 2, equation becomes which becomes 16-14-2 = 0 which becomes 2-2 = 0 which becomes 0 = 0 confirming x = 2 is good.
when x = -.25, equation becomes which becomes .25 + 1.75 - 2 = 0 which becomes 2 - 2 = 0 which becomes 0 = 0 confirming x = -.25 is good.
both answers are good.
your answers are:
x = 2 or x = -.25
bottom:
the equation you want to complete the squares on is:
the coefficient of the x^2 term has to be 1 in order for this method to work.
we divided both sides of the equation by 4 to make that happen and were left with:
completing the squares methods requires we take the square root of where k is a constant.
we do that by taking one half of the b term which is the coefficient of x.
the constant term of k will always be equal to one half of the b term squared.
one half of (7/4) = (7/8)
since the b term is negative, our factor will be:
.
.
since we want to be equal to , we have to subtract (49/64) from to make it equal to
our equation of:
becomes equivalent to:
we add (49/64) to both sides of the equation to get:
resume from "your equation becomes"
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