Steps to Complete the Square
Completing the square is a technique used in algebra to convert a quadratic equation in the standard form, ax^2 + bx + c = 0, into a perfect square trinomial
Completing the square is a technique used in algebra to convert a quadratic equation in the standard form, ax^2 + bx + c = 0, into a perfect square trinomial. This is done by adding or subtracting a constant term from both sides of the equation.
Here are the steps to complete the square:
Step 1: Make sure the coefficient of x^2 term, a, is equal to 1. If the coefficient is not 1, divide the entire equation by a to get it in the form x^2 + (b/a)x + c/a = 0.
Step 2: Move the constant term, c/a, to the other side of the equation, so the equation becomes x^2 + (b/a)x = -c/a.
Step 3: To complete the square, divide the coefficient of x, (b/a), by 2 and square that result [(b/2a)^2].
Step 4: Add the result obtained in step 3 to both sides of the equation. The left side of the equation will now be a perfect square trinomial.
Step 5: Simplify the left side of the equation by factoring it into a binomial squared.
Step 6: Take the square root of both sides of the equation to solve for x. Remember to consider both the positive and negative square root.
Step 7: Solve for x and simplify if necessary.
Let’s go through an example to illustrate these steps:
Example: Solve the quadratic equation x^2 + 6x + 9 = 0 by completing the square.
Step 1: We can see that the coefficient of x^2 is already 1, so we can move on to step 2.
Step 2: Move the constant term, 9, to the other side of the equation: x^2 + 6x = -9.
Step 3: Divide the coefficient of x, 6, by 2 and square it: (6/2)^2 = 9.
Step 4: Add 9 to both sides of the equation: x^2 + 6x + 9 = -9 + 9 -> x^2 + 6x + 9 = 0.
Step 5: Simplify the left side by factoring it into a binomial squared: (x + 3)^2 = 0.
Step 6: Take the square root of both sides: x + 3 = ± √0, which simplifies to x + 3 = 0.
Step 7: Solve for x: x = -3.
Therefore, the solution to the quadratic equation x^2 + 6x + 9 = 0 is x = -3.
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