Algebra QUADRATIC EQUATIONSFactoringQuadraticEquationsCertain complete quadratic equations can be solved by factoring. If the left-hand side of thegeneral form of a quadratic equation can be factored, the only way for the factored equation tobe true is for one or both of the factors to be zero. For example, the left-hand side of thequadratic equation x2 + x - 6 = 0 can be factored into (x + 3)(x - 2). The only way for theequation (x + 3) (x - 2) = 0 to be true is for either (x + 3) or (x - 2) to be zero. Thus, the rootsof quadratic equations which can be factored can be found by setting each of the factors equalto zero and solving the resulting linear equations. Thus, the roots of (x + 3)(x - 2) = 0 are foundby setting x + 3 and x - 2 equal to zero. The roots are x = -3 and x = 2.Factoring estimates can be made on the basis that it is the reverse of multiplication. Forexample, if we have two expressions (dx + c) and (cx + g) and multiply them, we obtain (usingthe distribution laws)(dx + c) (fx + g) = (dx) (fx) + (dx) (g) + (c) (fx) + cg == dfx2 + (dg + cf)x + cg.Thus, a statement (dx + c) (fx + g) = 0 can be writtendf x2+ (dg + cf)x + cg = 0.Now, if one is given an equation ax2 + bx + c = 0, he knows that the symbol a is the productof two numbers (df) and c is also the product of two numbers. For the example 3x2 - 4x - 4 =0, it is a reasonable guess that the numbers multiplying x2 in the two factors are 3 and 1,although they might be 1.5 and 2. The last -4 (c in the general equation) is the product of twonumbers (eg), perhaps -2 and 2 or -1 and 4. These combinations are tried to see which gives theproper value of b (dg + ef), from above.There are four steps used in solving quadratic equations by factoring.Step 1. Using the addition and subtraction axioms, arrange the equation in thegeneral quadratic form ax2 + bx + c = 0.Step 2. Factor the left-hand side of the equation.Step 3. Set each factor equal to zero and solve the resulting linear equations.Step 4. Check the roots.Rev. 0 Page 21 MA-02
Integrated Publishing, Inc. - A (SDVOSB) Service Disabled Veteran Owned Small Business