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solve by completing the square

by Becky











































a quadratic equation has the general form:

ax² + bx + c = 0

The equation can be solved by reformatting it so that the constant "c" is on the right side of the equation, and the left side of the equation is written as a perfect square.

This is called "completing the square".


please me completing the problem.

3x^2 - 12x + 2 = 0

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Nov 29, 2010
Completing the Square
by: Staff

The question:

by Becky


please me completing the problem.

3x^2-12x+2=0

The answer:

Completing the square is the process used in the derivation of the quadratic formula:

We are going to FORCE THE EQUATION TO LOOK LIKE THIS:

(x-number)^2 = another number

Once we reach that point, we will take the square root of each side of the equation:

Sqrt[(x-number)^2] = Sqrt(another number)

x – number = Sqrt(another number)

Adding + number to each side of the equation, we will get:

x – number + number = Sqrt(another number) + number

x = number ± Sqrt(another number)


That’s the plan. Now, let’s apply that approach to your equation:

1. Your initial equation
3x^2 - 12x + 2 = 0

2. Subtract 2 from each side of the equation to eliminate the +2 on the left hand side:
3x^2 - 12x + 2 - 2 = 0 – 2
3x^2 - 12x = – 2

3. Now we are going to eliminate coefficient of 3 from the 3x^2. To accomplish this, we will divide each side of the equation by 3:

3x^2 - 12x = – 2
(3x^2 - 12x)/3 = – 2/3
(3x^2)/3 – (12x)/3 = – 2/3
x^2 – 4x = – 2/3

4. Now we are going to add a number to both sides of the equation that will make the left hand side of the equation a perfect square:

x^2 – 4x = – 2/3

That number is the number +4. You can guess at it until you find it, or you can compute it by dividing the 4 in the second term by 2, then squaring the result:

4/2 = 2
2^2 = 4

The equation now looks like this

x^2 – 4x = – 2/3
x^2 – 4x + 4 = +4 – 2/3
x^2 – 4x + 4 = 3 & 1/3, or 10/3
x^2 – 4x + 4 = 10/3

When we factor the left side of the equation, we can see it is now a perfect square:

x^2 – 4x + 4 = 10/3
(x – 2)*(x – 2) = 10/3
(x – 2)^2 = 10/3

5. Now we are going take the square root of each side of the equation:

(x – 2)^2 = 10/3
Sqrt[(x – 2)^2] = Sqrt[10/3]
x – 2 = Sqrt[10/3]

6. One last step. Add +2 to each side of the equation:

x – 2 = Sqrt[10/3]
x – 2 + 2 = Sqrt[10/3] + 2
x = 2 ± Sqrt[10/3]

this is our final answer: x = 2 ± Sqrt[10/3]

our final answer can also be re-written: x = 2 ± Sqrt[30]/3

However, if you want to compute both numerical answers:

x = 2 ± Sqrt[10/3]
x = 2 ± 1.82574185835
x = 3.8257419 and x = 0.1742581



Thanks for writing.


Staff
www.solving-math-problems.com


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