![]() Let’s transpose the constant term to the other side of the equation: x 2 - 4x = 8. Using formula, ax 2 + bx + c = a(x + m) 2 + n. Finding the square,Įxample 2: Use completing the square formula to solve: x 2 - 4x - 8 = 0. Thus, from both methods, the term that should be added to make the given expression a perfect square trinomial is 49/4. Using the formula, the term that should be added to make the given expression a perfect square trinomial is, Here are a few examples of the application of completing the square formula.Įxample 1: Using completing the square formula, find the number that should be added to x 2 - 7x in order to make it a perfect square trinomial.Ĭomparing the given expression with ax 2 + bx + c, a = 1 b = -7 Let us study this in detail using illustrations in the following sections. These formulas are derived geometrically. To complete the square in the expression ax 2 + bx + c, first find the values of m and n using the above formulas and then substitute these values in: ax 2 + bx + c = a(x + m) 2 + n. 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. ![]() 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. ![]()
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