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quadratic equation

Please help me solve the equation with the folowing method:  x2+12x-64=0

 

1.  Move the constant term to the right side of the equation.

2.  Multiply each term in the equation by four times the coefficient of the x2 term.

3.  Square the coefficient of the original x term and add it to both sides of the equation.

4.  Take the square root of both sides.

5.  Set the left side of the equation equal to the positive square root of the number on the right side and solve for x.

6.  Set the left side of the equation equal to the negative square root of the number on the right side of the equation and solve for x.

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4 Answers

The method that you have given is similar to completing the square, but it has variations that I am not familiar with.  I will do my best to explain what is meant by each step along with showing what the equation should look like

We begin with x2 + 12x - 64 = 0

1.  The first step says to "move" the constant term to the right side of the equation.  This means that we will add 64 for to each side so that there will be a constant on the right side, but no constant on the left. 

The new equation will be x2 + 12x = 64.

2.  The second step says to multiply each term in the equation by four times the coefficient of the x2 .  The coefficient of the x2 term is 1, so we are going to multiply each term by 1 times 4.  1 times 4 is 4, so we will multiply each term by 4.

The new equation will be 4x2 + 48x = 256.

3.  The third step says that we need to square the original coefficient of the x term and add it to both sides.  The original coefficient of the x term is 12.  When I square 12, I get 144.  I need to add 144 to both sides.  I then need to simplify the right side (add 256 and 144 to get 400)

The new equation will be 4x2 + 48x + 144 = 400

4.  The fourth step says to take the square root of both sides.  Before you do this, you should factor the left side into a square term.  This means that you will come up with two identical factors for the term on the left.  The coefficient of the x term of the factor will be the square root of your x2 factor, while the constant term of the factor will be the square root of your constant.  In this case, the coefficient on the x term of the factor will be 2 (the square root of 4) and the constant term will be 12 (the square root of 144).

The factored version from step 3 will be (2x + 12)2 = 400.  If you want to double check this, foil the two factors and see if you get the correct quadratic.

After you find the square factor of the quadratic on the left, take the square root of both sides, but remember that you want the positive and negative square roots on the right.  Your left hand term will be the factor inside parenthesis.

The new equations (there are two of them now) are 2x + 12 = 20 and 2x + 12 = -20.

5.  We are going to solve the first equation in this step.  Solve 2x + 12 = 20 by subtracting 12 from each side and then dividing by 2.  The answer should be 4.

6.  Finish by solving 2x + 12 = -20.  First subtract 12 from each side, then divide each side by 2.  The answer will be -16.

 

Comments

x2+12x-64=0

This is called completing the square.

x2+12x=64

(Move the constant term to the right side of the equation.)

4(x2+12x)=4(64)

(Multiply each term in the equation by four times the coefficient of the x2 term.)

4x2+48x=256

4x2+48x+12(12)=256+12(12)

(Square the coefficient of the original x term and add it to both sides of the equation.)

4x2+48x+144=256+144 <---The left side of this is supposed to factor easily as a perfect square:

(2x2+12)2=400

√((2x+12)2)=√(400) AND √((2x+12)2)=-√(400)

(Take the square root of both sides. Set the left side of the equation equal to the positive square root of the number on the right side and solve for x. Set the left side of the equation equal to the negative square root of the number on the right side of the equation and solve for x.)

2x+12 =20 AND 2x+12 =-20

2x+12 =20 AND 2x+12 =-20

2x=8 AND 2x=-32

x=4 AND x = -16

Hi, looks like this is one version of completing the square.

Original equation: x2 + 12x - 64 = 0

1) x2 + 12x = 64

2) The coefficient of the x2 term is 1, so we multiply each term by 4.

-> 4x2 + 46x = 256

3) The original x term is 12, so we add to both sides 122 = 144.

-> 4x2 + 46x + 144 = 256 + 144

4) To take the square root of the left side, it's best we simplify a bit.

-> 4 (x2 + 12x + 36) = 400

-> 4 (x + 6)2 = 400

Now we can take the square root.

-> 2 (x + 6) = 20

5) 2 (x + 6) = 20

-> x + 6 = 10

-> x = 4

6) 2 (x + 6) = -20

-> x + 6 = -10

-> x = -16