solve quadratic equations by completing the square

Solve quadratic equations by completing the square how to#

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.

Solve quadratic equations by completing the square how to#

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.