![]() ![]() Instead of using the complex step-wise method for completing the square, we can use the following simple formula to complete the square. The formula for completing the square is: ax 2 + bx + c ⇒ a(x + m) 2 + n, where Note: Completing the square formula is used to derive the quadratic formula.Ĭompleting the square formula is a technique or method that can also be used to find the roots of the given quadratic equations, ax 2 + bx + c = 0, where a, b and c are any real numbers but a ≠ 0. A quadratic expression in variable x: ax 2 + bx + c, where a, b and c are any real numbers but a ≠ 0, can be converted into a perfect square with some additional constant by using completing the square formula or technique. Add and subtract (b/2a) 2 after the 'x' term and simplify.Ĭompleting the square formula is a technique or method to convert a quadratic polynomial or equation into a perfect square with some additional constant. ![]() Note: To complete the square in an expression ax 2 + bx + c Step 5: Simplify the last two numbers.Step 4: Factorize the perfect square trinomial formed by the first 3 terms using the identity x 2 + 2xy + y 2 = (x + y) 2.Step 3: Add and subtract the above number after the x term in the expression whose coefficient of x 2 is 1.Step 2: Find the square of the above number.Step 1: Find half of the coefficient of x.If the coefficient of x 2 is NOT 1, we will place the number outside as a common factor. Now that we have gone through the steps of completing the square in the above section, let us learn how to apply the completing the square method using an example.Įxample: Complete the square in the expression -4x 2 - 8x - 12.įirst, we should make sure that the coefficient of x 2 is '1'. How to Apply Completing the Square Method? Step 5: Factorize the polynomial and apply the algebraic identity x 2 + 2xy + y 2 = (x + y) 2 (or) x 2 - 2xy + y 2 = (x - y) 2 to complete the square.Step 4: Add and subtract the square obtained in step 2 to the x 2 term.Step 3: Take the square of the number obtained in step 1.Step 2: Determine half of the coefficient of x.Step 1: Write the quadratic equation as x 2 + bx + c.Given below is the process of completing the square stepwise: To apply the method of completing the square, we will follow a certain set of steps. But, how do we complete the square? Let us understand the concept in detail in the following sections. Since we have (x + m) whole squared, we say that we have "completed the square" here. In such cases, we write it in the form a(x + m) 2 + n by completing the square. X 2 + 2x + 3 cannot be factorized as we cannot find two numbers whose sum is 2 and whose product is 3. Let us have a look at the following example to understand this case. But sometimes, factorizing the quadratic expression ax 2 + bx + c is complex or NOT possible. We know that a quadratic equation of the form ax 2 + bx + c = 0 can be solved by the factorization method. If you get stuck on the fractions, the right-hand term in the parentheses will be half of the x-term.The most common application of completing the square method is factorizing a quadratic equation, and henceforth finding the roots or zeros of a quadratic polynomial or a quadratic equation. We especially designed this trinomial to be a perfect square so that this step would work: Now rewrite the perfect square trinomial as the square of the two binomial factors That is 5/2 which is 25/4 when it is squared Now we complete the square by dividing the x-term by 2 and adding the square of that to both sides of the equation. X² + 5x = 3/4 → I prefer this way of doing it Or, you can divide EVERY term by 4 to get ĭivide through the x² term and x term by 4 to factor it out So, we have to divide the x² AND the x terms by 4 to bring the coefficient of x² down to 1. In the example following rule 2 that we were supposed to try, the coefficient of x² is 4. As shown in rule 2, you have to divide by the value of a (which is 4 in your case). You are correct that you cannot get rid of it by adding or subtracting it out. This would be the same as rule 2 (and everything after that) in the article above. ![]()
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