Question 177853
equation # 29 is: ******************************************************
{{{2*x^2 - 6*x + 1 = 0}}}
standard form of the equation is:
ax^2 + bx + c = 0
this is already in standard form.
a term = 2
b term = (-6)
c term = 1
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first you move the constant to the right side of the equation by subtracting 1 from both sides of the equation to get :
{{{2*x^2 - 6*x = -1}}}
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you then divide both sides of the equation by 2 to get:
{{{x^2 - 3*x = (-1/2)}}}
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you then take half of 3 and factor the left side of the equation to get:
{{{(x - (3/2))^2 - (3/2)^2 = (-(1/2))}}}
this takes a little explaining.
start of explanation.
here's an example (not anything to do with this problem because the numbers are changed to make it simple).
take {{{x^2 + 2x}}}.
if you take half the 2 and make this equal to {{{(x+1)^2}}}, the answer will be:
{{{x^2 + 2x + (1)^2}}}
that {{{(1)^2}}} is extra, so you have to subtract it to keep the original equality intact.
you get:
{{{(x+1)^2 - 1 = (x^2 + 2x + 1)- 1 = x^2 + 2x}}} which is what you started off with.
this is exactly what we did above:
we took {{{x^2 - 3*x}}} and factored it to get:
{{{(x-3/2)^2 - (3/2)^2}}}
if you do the multiplication, you will see that:
{{{(x-3/2)^2 - (3/2)^2 = x^2 - 3x + (3/2)^2 - (3/2)^2 = x^2 - 3x}}}
end of explanation.
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you then add the (3/2)^2 term to both sides of the equation to get:
{{{(x - (3/2))^2 = (-(1/2)) + (3/2)^2}}}
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you then take the square root of both sides of the equation to get:
{{{x - (3/2)}}} = +/- {{{sqrt((-1/2) + (3/2)^2)}}}
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you then add ((3/2)) to both sides of the equation to get:
x = +/- {{{(sqrt((-1/2) + (3/2)^2)) + (3/2)}}}
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after doing the math (i used a calculator), you will get:
x = 2.8228...
or
x = .1771...
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to prove these values are correct, substitute them in the original equation (again i used the calculator with the full rather than the truncated values)
i took the original equation of 2x^2 - 6x + 1 = 0
and substituted these values to get:
0 = 0 both times proving both values are good.
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equation number 32 is: ***********************************************
{{{-4*x^2 + 8*x - 3 = 0}}}
standard form of the equation is:
ax^2 + bx + c = 0
this is already in standard form.
a term = -4
b term = 8
c term = -3
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first you move the constant to the right side of the equation by adding 3 to both sides of the equation to get :
{{{-4*x^2 + 8*x = 3}}}
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you then divide both sides of the equation by (-4) to get:
{{{x^2 - 2*x = -3/4}}}
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you then take half of 2 and factor the left side of the equation to get:
{{{(x - 1)^2 - (1)^2 = -3/4)}}}
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you then add the (1)^2 term to both sides of the equation to get:
{{{(x-1)^2 = (-3/4) + 1}}}
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you then take the square root of both sides of the equation to get:
{{{x-1}}} = +/- {{{sqrt((-3/4) + 1)}}}
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you then add 1 to both sides of the equation to get:
x = +/- {{{(sqrt((-3/4) + 1)) + 1}}}
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after doing the math (i used a calculator), you will get:
x = 1.5
or
x = .5
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to prove these values are correct, substitute them in the original equation (again i used the calculator with the full rather than the truncated values)
i took the original equation of -4x^2 + 8x - 3 = 0
and substituted these values to get:
0 = 0 both times proving both values are good.
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i will do number 30 next and i will leave number 31 for you to do.
if you follow the steps and understand what is going on, you should be able to complete it.
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equation number 30 is: **************************************************
{{{-x^2 - 8*x + 5 = 0}}}
standard form of the equation is:
ax^2 + bx + c = 0
this is already in standard form.
a term is -1.
b term is -8.
c term is 5
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first you move the constant to the right side of the equation by subtracting 5 from both sides of the equation to get :
{{{-x^2 - 8*x = -5}}}
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you then divide both sides of the equation by -1 to get:
{{{x^2 + 8*x = 5}}}
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you then take half of 8 and factor the left side of the equation to get:
{{{(x + 4)^2 - 4^2 = 5}}}
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you then add the 4^2 term to both sides of the equation to get:
{{{(x+4)^2 = 5 + 4^2}}}
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you then take the square root of both sides of the equation to get:
{{{x+4 = sqrt(5 + 4^2)}}}
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you then subtract 4 from both sides of the equation to get:
x = +/- {{{sqrt(5 + 4^2)-4}}}
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after doing the math (i used a calculator), you will get:
x = .5828...
or
x = =-8.5825...
---
to prove these values are correct, substitute them in the original equation (again i used the calculator with the full rather than the truncated values)
i took the original equation of -x^2 - 8x + 5 = 0 
and substituted these values to get:
0 = 0 both times proving both values are good.
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by now, you should be able to do number 31 by yourself.
let me know if you are having problems.