Question 896495: Toni is solving this equation by completing the square.
ax^2 + bx + c = 0 (where a is greater than or equal to zero)
Step one: ax^2 + bx = -c
Step two: x^2 + b/ax=-c/a
Step 3: ?
A. x^2=-c/b - b/ax
B. x + b/a= c/ax
C. x^2 + b/ax + b/2a = -c/a +b/2a
D. x^2 + b/ax + (b/2a)^2 =-c/a + (b/2a)^2
Found 3 solutions by nerdybill, rothauserc, Theo:
Answer by nerdybill(7384)   (Show Source):
Answer by rothauserc(4718)  (Show Source):
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! i believe your solution is option D.
start with ax^2 + bx + c = 0
divide both sides by a to get:
x^2 + (b/a)x + (c/a) = 0
subtract (c/a) from both sides to get:
x^2 + (b/a)x = -(c/a)
now take half the coefficient of the x term and square it and add it to both sides of the equation to get:
x^2 + (b/a)x + (b/2a)^2 = -(c/a) + (b/2a)^2
this is selection D which is your answer.
what comes next though?
x^2 + (b/a)x + (b/2a)^2 is a perfect square and can be factored to get you:
(x + (b/2a)^2 = -(c/a) + (b/2a)^2
when you square (x + (b/2a))^2, you get:
(x + (b/2a) * (x + (b/2a) which is equal to:
x^2 + (b/2a)^2 + (b/2a)x + (b/2a)x which simplifies to:
x^2 + 2 * (b/2a)x + (b/2a)^2 which further simplifies to:
x^2 + (b/a)x + (b/2a)^2
it's easier to see this with numbers.
assume your equation is 2x^2 + 8x + 8 = 0 and you want to factor it using the completing the square method.
step 1:
divide both sides of the equation by 2 to get;
x^2 + 4x + 4 = 0
step 2:
subtract 4 from both sides of the equation to get:
x^2 + 4x = -4
take half the coefficient of the x term to get 2 and square it to get 4 and add it to both sides of the equation to get:
x^2 + 4x + 4 = -4 + 4 *****
***** this is your selection D with numbers instead of letters.
simplify this to get:
x^2 + 4x + 4 = 0
now factor the equation on the left, which is a perfect square.
you will get:
(x + 2)^2 = 0
now take the square root of both sides of this equation to get:
x + 2 = 0
now solve for x to get x = -2
the equation has been factored and the solution is x = -2
the equation of 2x^2 + 8x + 8 = 0 will be true when x = -2
replace x with -2 and you get:
2*(-2)^2 + 8*(-2) + 8 = 0 which becomes:
8 - 16 + 8 = 0 which becomes 0 = 0 which confirms the solution is correct.
selection D is your next step.
if you take any equation in the form of x^2 + (b/a)x = -(c/a) and you take half the coefficient of the x term and square it and then add it to both sides of the equation, you form the perfect square on the left side of the equation.
x^2 + (b/a)x + (b/2a)^2 is a perfect square because, when you factor it, you get a factor that, when squared, it equal to it.
x^2 + (b/a)x + (b/2a)^2 is equal to (x + (b/2a))^2
you do have to add (b/2a)^2 to both sides of the equation, however, in order to preserve the equality.
another example that is not as clean, but is still accurate according to the method.
your equation is 3x^2 + 15x - 32 = 0
you want to factor this using the completing thed square method.
divide both sides of the equation by 3 to get:
x^2 + (15/3)x - (32/3) = 0
add (32/3) to both sides of the equation to get:
x^2 + (15/3)x = (32/3)
simplify to get:
x^2 + 5x = (32/3)
take half the coefficient of the x term and square it and add it to both sides of the eqution to get:
x^2 + 5/x + (5/2)^2 = (32/3) + (5/2)^2
the left side of the equation is now a perfect square and can be factored to get:
(x + (5/2))^2 = (32/3) + (5/2)^2
notice that the constant term in the factor is (5/2)
it's the same constant that you squared and then added to both sides of the eqaution.
if you simplify (x + 5/2)^2, you get (x + (5/2)) * (x + (5/2)) is equal to:
x^2 + (5/x)^2 + (5/2)x + (5/2)x which simplifies to:
x^2 + 5x + (5/2)^2
there's your perfect square again.
here's a link that talks about how to factor using the completing the square method.
http://www.purplemath.com/modules/solvquad3.htm
that link includes other methods as well which are worth going through if you are interested in learning more about how to factor quadratics.
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