Question 202134: Hello,
I need some help on quadratic equations. Any and all help will be appreciated.
1. Solve using the quadratic formula: x^2 - 3x = 4x -1
2. Solve using the quadratic formula: x^2 -5x - 1 = -7
3. Solve by completing the square: x^2 + 8x + 2 = 0
Thank you in advance for your help.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the quadratic formula is:
x = 
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the general form of the quadratic equation is:
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to use the quadratic equation, you find the values for a, b, and c, and plug them into the quadratic equation and solve.
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your first problem is:
first you need to convert it to the standard form.
subtract 4x from both sides of the equation and add 1 to both sides of the equation to get:
a = 1
b = -7
c = 1
plug these values into the quadratic equation to get:
x = 
which becomes:
x = 
this answer works as is but you can reduce it further by simplifying the number under the square root sign as follows:
x = 
which reduces to:
x = 
all you need to do now is verify the answer is correct by plugging those values for x into the original equation to see if the equation is true. I cheated by using my calculator, but the answers are verified to be true so these answers are good.
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your second problem is:
x^2 -5x - 1 = -7
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you do the same thing.
convert it to standard form.
get the values of a, b, and c.
plug them into the formulas and solve.
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i'll solve without explanation this time and you can follow along or try to solve it on your own.
x^2 - 5x - 1 = -7
add 7 to both sides to get:
x^2 - 5x + 6 = 0
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a = 1
b = -5
c = 6
x =
x = 
which becomes
x = 
which becomes
x = 
which becomes
x = 
which becomes
x = 3 or x = 2
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plug these values into the original equations to see if the equations are true.
original equation is:
x^2 -5x - 1 = -7
when x = 2 this becomes
4 - 10 - 1 = -7
which becomes
-6-1 = -7 which is true.
when x = 3 this becomes
9 - 15 - 1 = -7
which becomes
-6 -1 = -7 which is also true.
the answers are good.
x =
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your last equation needs to be solved by completing the squares.
this is a different technique but gets you to the same answer.
you could also use the quadratic equation here as well, but sometimes completing the square is easier.
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your last equation is:
you would subtract 2 from both sides of the equation to get:
you take 8/2 = 4 and use that value to complete the square.
you take  to get:
 .
so what you have is:
 .
if you subtract 16 from that, you have:
you can now go back to your original equation after modification which was:
and substitute  for  to get:
you add 16 to both sides of the equation to get:
you have just completed the square.
solving by taking the square root of both sides and you will get:
 = +/- 
which becomes:
you verify that this answer is valid by plugging it into the original equation which was:
the values check out ok so the answer is good.
note that if you had solved this equation using the quadratic formula, you would have gotten the same answer.
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note that the standard form of the quadratic equation is:
 .
to complete the square, you need to move the c over to the right side and you need to divide both sides of the equation by a, if a is not already 1.
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you will wind up with:
before dividing by a if it is not 1 to start with.
if a is not one, you will wind up with:
which is the same as:
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i'll show you how that works with a simple example.
once you have the standard form for completing the square, the operations are the same as above.
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take 
this is the standard form of the quadratic equation of .
note that a is not = to 1
first step:
move the c over to the right hand side by subtracting it from both sides of the equation.
you get:
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since the a coefficient is not 1 (it's 3), you need to divide both sides of the equation by 3 to make it 1.
it becomes:
which becomes:
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now you're in the standard form for completing the square.
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take 1/2 the b factor which means take 1/2 of 3.
you get 3/2
your factor that will be squared is  .
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you get:
your left side of the equation was 
this equals 
your equation which was:
becomes:
you add (9/4) to both sides to get:
this becomes:
which becomes:
take the square root of both sides to get:
 = +/- 
this means:
or
or
x = both.
you wind up with:
x = -1 or x = -2 or both.
both values of x checked out in the original equation so they're both good.
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