# How to solve quadratic equation by completing the square

2
by munoy

2016-07-09T20:14:40+08:00
We can complete the square to solve aQuadratic Equation (find where it is equal to zero).

But a general Quadratic Equation can have acoefficient of a in front of x2:

ax2 + bx + c = 0

But that is easy to deal with ... just divide the whole equation by "a" first, then carry on:

x2 + (b/a)x + c/a = 0

Steps

Now we can solve a Quadratic Equation in 5 steps:

Step 1 Divide all terms by a (the coefficient of x2).
Step 2 Move the number term (c/a) to the right side of the equation.
Step 3 Complete the square on the left side of the equation and balance this by adding the same value to the right side of the equation.

We now have something that looks like (x + p)2 = q, which can be solved rather easily:

Step 4 Take the square root on both sides of the equation.Step 5 Subtract the number that remains on the left side of the equation to find x.

Examples

Here are two examples:

Example 1: Solve x2 + 4x + 1 = 0

Step 1 can be skipped in this example since the coefficient of x2 is 1

Step 2 Move the number term to the right side of the equation:

x2 + 4x = -1

Step 3 Complete the square on the left side of the equation and balance this by adding the same number to the right side of the equation.

(b/2)2 = (4/2)2 = 22 = 4

x2 + 4x + 4 = -1 + 4

(x + 2)2 = 3

Step 4 Take the square root on both sides of the equation:

x + 2 = ±√3 = ±1.73 (to 2 decimals)

Step 5 Subtract 2 from both sides:

x = ±1.73 – 2 = -3.73 or -0.27
2016-07-09T20:26:03+08:00
In completing the square,
1. Arrange the terms in standard form.
2. Divide all terms by the coefficient of x^2 (the value of a).
3. Move the constant (the value of c) to the right side of the equation.
4. Divide the 2nd term by 2 and square the answer.
5. The answer will be your 3rd term. Add also the answer to the right side of the equation.
6. Copy the first term, copy the sign of the second term and the square root of the third term. For example ---> k^2 - 6k + 9 ----> (k-3)^2
7. To cancel the square, square root both terms
8. The answer must have ± sign
9. Then, find the value of the variable. For example, k-3 =
±3 --> k=3+3 --> k=6      k-3 = -3 + 3  k=0

You must solve for both signs ±