![]() 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 Thus 3x 2 + 6x = 0 is factorized as 3x(x + 2) = 0.The algebraic common factor is x in both terms.The numerical factor is 3 (coefficient of x 2) in both terms.Let us solve an example to understand the factoring quadratic equations by taking the GCD out.Ĭonsider this quadratic equation: 3x 2 + 6x = 0 Using Algebraic Identities (Completing the Squares)įactoring Quadratics by Taking Out The GCDįactoring quadratics can be done by finding the common numeric factor and the algebraic factors shared by the terms in the quadratic equation and then take them out.There are different methods that can be used for factoring quadratic equations. Thus the equation has 2 factors (x+3) and (x-3)įactoring quadratics gives us the roots of the quadratic equation. Verify by substituting the roots in the given equation and check if the value equals 0. Thus the equation has 2 factors (x + 3) and (x + 2)ģ and -3 are the two roots of the equation. Consider the quadratic equation x 2 + 5x + 6 = 0 Let us go through some examples of factoring quadratics:ġ. Hence, factoring quadratics is a method of expressing the quadratic equations as a product of its linear factors, that is, f(x) = (x - \(\alpha\))(x - \(\beta\)). Thus, (x - \(\beta\)) should be a factor of f(x). 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. Quadratics which arise from observed measurements and experimental results are more likely to need the use of the quadratic formula for solving.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. The quadratic formula may also demonstrate when no solution exists (something that is difficult to see with factoring). If the coefficient of #x^2# and the coefficient with no #x# element have relatively few factors, time invested in attempting to factor the quadratic is usually worthwhile.Īlso if you know the source of the quadratic, you can sometimes guess if factoring is likely to be successful (for example if it is a simple mathematical model of a situation or a question developed by a friendly math teacher).Īnd you have to be careful with arithmeticīut it will give results when factoring won't work. Quadratics with coefficients that involve roots would be one example of "ugly".Īfter that (if the "ugly" rule doesn't apply):įactoring is usually faster and less prone to arithmetic mistakes (if you are working by hand). If the quadratic looks particularly " ugly " use the quadratic formula. The following is mostly some rules of thumb. This is actually a very good question, but not one with a really definitive answer.
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