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. Step 5: Substitute either value (we'll use `+4`) into the `u` bracket expressions, giving us the same roots of the quadratic equation that we found above:įor more on this approach, see: A Different Way to Solve Quadratic Equations (video by Po-Shen Loh).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. Step 3: Set that expansion equal to the constant term: `1 - u^2 = -15` Step 1: Take −1/2 times the x coefficient. The following approach takes the guesswork out of the factoring step, and is similar to what we'll be doing next, in Completing the Square. We could have proceded as follows to solve this quadratic equation. (Similarly, when we substitute `x = -3`, we also get `0`.) Alternate method (Po-Shen Loh's approach) We check the roots in the original equation by Now, if either of the terms ( x − 5) or ( x + 3) is 0, the product is zero. (v) Check the solutions in the original equation (iv) Solve the resulting linear equations (i) Bring all terms to the left and simplify, leaving zero on Using the fact that a product is zero if any of its factors is zero we follow these steps: If you need a reminder on how to factor, go back to the section on: Factoring Trinomials. Solving a Quadratic Equation by Factoringįor the time being, we shall deal only with quadratic equations that can be factored (factorised). This can be seen by substituting x = 3 in the The quadratic equation x 2 − 6 x + 9 = 0 has double roots of x = 3 (both roots are the same) In this example, the roots are real and distinct. This can be seen by substituting in the equation: (We'll show below how to find these roots.) The quadratic equation x 2 − 7 x + 10 = 0 has roots of The solution of an equation consists of all numbers (roots) which make the equation true.Īll quadratic equations have 2 solutions (ie. x 3 − x 2 − 5 = 0 is NOT a quadratic equation because there is an x 3 term (not allowed in quadratic equations).bx − 6 = 0 is NOT a quadratic equation because there is no x 2 term.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |