SOLUTION: Please show me how to Solve the equation x2 + 8x � 2 = 0 using both The quadratic formula and Completing the square.

Question 389186: Please show me how to Solve the equation x2 + 8x � 2 = 0 using both The quadratic formula and Completing the square.

Found 2 solutions by haileytucki, jim_thompson5910:
Answer by haileytucki(390) (Show Source): You can put this solution on YOUR website!
Quadratic First (~=square root)
x^(2)+8x-2=0
Use the quadratic formula to find the solutions. In this case, the values are a=1, b=8, and c=-2.
x=(-b\~(b^(2)-4ac))/(2a) where ax^(2)+bx+c=0
Use the standard form of the equation to find a, b, and c for this quadratic.
a=1, b=8, and c=-2
Substitute in the values of a=1, b=8, and c=-2.
x=(-8\~((8)^(2)-4(1)(-2)))/(2(1))
Simplify the section inside the radical.
x=(-8\6~(2))/(2(1))
Simplify the denominator of the quadratic formula.
x=(-8\6~(2))/(2)
First, solve the + portion of +-.
x=(-8+6~(2))/(2)
Factor out the GCF of 2 from each term in the polynomial.
x=(2(-4)+2(3~(2)))/(2)
Factor out the GCF of 2 from -8+6~(2).
x=(2(-4+3~(2)))/(2)
Reduce the expression (2(-4+3~(2)))/(2) by removing a factor of 2 from the numerator and denominator.
x=(-4+3~(2))
Remove the parentheses around the expression -4+3~(2).
x=-4+3~(2)
Next, solve the - portion of +-.
x=(-8-6~(2))/(2)
Factor out the GCF of 2 from each term in the polynomial.
x=(2(-4)+2(-3~(2)))/(2)
Factor out the GCF of 2 from -8-6~(2).
x=(2(-4-3~(2)))/(2)
Reduce the expression (2(-4-3~(2)))/(2) by removing a factor of 2 from the numerator and denominator.
x=(-4-3~(2))
Remove the parentheses around the expression -4-3~(2).
x=-4-3~(2)
The final answer is the combination of both solutions.
x=-4+3~(2),-4-3~(2)_x=0.2426407,-8.24264

Now, completing the square

x^(2)+8x-2=0
Since -2 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 2 to both sides.
x^(2)+8x=2
To create a trinomial square on the left-hand side of the equation, add a value to both sides of the equation that is equal to the square of half the coefficient of x. In this problem, add (4)^(2) to both sides of the equation.
x^(2)+8x+16=2+16
Add 16 to 2 to get 18.
x^(2)+8x+16=18
Factor the perfect trinomial square into (x+4)^(2).
(x+4)^(2)=18
Take the square root of each side of the equation to setup the solution for x.
~((x+4)^(2))=\~(18)
Remove the perfect root factor (x+4) under the radical to solve for x.
(x+4)=\~(18)
Pull all perfect square roots out from under the radical. In this case, remove the 3 because it is a perfect square.
(x+4)=\3~(2)
First, substitute in the + portion of the \ to find the first solution.
(x+4)=3~(2)
Remove the parentheses around the expression x+4.
x+4=3~(2)
Since 4 does not contain the variable to solve for, move it to the right-hand side of the equation by subtracting 4 from both sides.
x=-4+3~(2)
Move all terms not containing x to the right-hand side of the equation.
x=3~(2)-4
Next, substitute in the - portion of the \ to find the second solution.
(x+4)=-3~(2)
Remove the parentheses around the expression x+4.
x+4=-3~(2)
Since 4 does not contain the variable to solve for, move it to the right-hand side of the equation by subtracting 4 from both sides.
x=-4+-3~(2)
Move all terms not containing x to the right-hand side of the equation.
x=-3~(2)-4
The complete solution is the result of both the + and - portions of the solution.
x=3~(2)-4,-3~(2)-4
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
Solving by use of the quadratic formula:

Start with the given equation.

Notice that the quadratic is in the form of where , , and

Let's use the quadratic formula to solve for "x":

Start with the quadratic formula

Plug in , , and

Square to get .

Multiply to get

Rewrite as

Add to to get

Multiply and to get .

Simplify the square root (note: If you need help with simplifying square roots, check out this solver)

Break up the fraction.

Reduce.

or Break up the expression.

So the solutions are or

----------------------------------------------------------------------------

Solving by use of completing the square:

First we need to complete the square for the expression

Start with the given expression.

Take half of the coefficient to get . In other words, .

Now square to get . In other words,

Now add and subtract . Make sure to place this after the "x" term. Notice how . So the expression is not changed.

Group the first three terms.

Factor to get .

Combine like terms.

So after completing the square, transforms to . So .

So is equivalent to .

Now let's solve

Start with the given equation.

Add to both sides.

Combine like terms.

Take the square root of both sides.

or Break up the "plus/minus" to form two equations.

or Simplify the square root.

or Subtract from both sides.

--------------------------------------

Answer:

So the solutions are or .

Notice how we get the same answers. So either method works.

If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim
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