Step 3: Add the square of (b/2a) to both the sides of quadratic equation x 2 + (b/a) x = -c/a.Obtained equation is x 2 + (b/a) x = -c/a Step 2: Subtract c/a from both the sides of quadratic equation x 2 + (b/a) x + c/a = 0.Now, the obtained equation is x 2 + (b/a) x + c/a = 0 Step 1: Divide both the sides of quadratic equation ax 2 + bx + c = 0 by a.Steps to factorize quadratic equation ax 2 + bx + c = 0 using completeing the squares method are: The main algebraic identities which are used for completing the squares are: The process of factoring quadratics can be done by completing the squares which require the use of algebraic identities. Thus, (x + 2) and (x + 6) are the factors of x 2 + 8x + 12 = 0 Taking the common factor (x + 6) out, we have Hence, we split the middle term and write the quadratic equation as: Now, we can see that the factor pair (2, 6) satisfies our purpose as the sum of 6 and 2 is 8 and the product is 12. Split the middle term 8x in such a way that the factors of the product of 1 and 12 add up to make 8. We determine the factor pairs of the product of a and c such that their sum is equal to b. We split the middle term b of the quadratic equation ax 2 + bx + c = 0 when we try to factorize quadratic equations. The product of the roots in the quadratic equation ax 2 + bx + c = 0 is given by \(\alpha\beta\) = c/a.The sum of the roots of the quadratic equation ax 2 + bx + c = 0 is given by \(\alpha + \beta\) = -b/a.
Splitting the Middle Term for Factoring Quadratics
Similarly, if x = \(\beta\) is the second root of f(x) = 0, then x = \(\beta\) is a zero of f(x). Thus, (x - \(\alpha\)) should be a factor of f(x). This means that x = \(\alpha\) is a zero of the quadratic expression f(x). Suppose that x = \(\alpha\) is one root of this equation. Consider a quadratic equation f(x) = 0, where f(x) is a polynomial of degree 2. They are the zeros of the quadratic equation. Every quadratic equation has two roots, say \(\alpha\) and \(\beta\). The factor theorem relates the linear factors and the zeros of any polynomial. Factorization of quadratic equations can be done using different methods such as splitting the middle term, using the quadratic formula, completing the squares, etc. This method is also is called the method of factorization of quadratic equations. Factoring quadratics is a method of expressing the quadratic equation ax 2 + bx + c = 0 as a product of its linear factors as (x - k)(x - h), where h, k are the roots of the quadratic equation ax 2 + bx + c = 0.