Factoring Quadratic Equations:
Certain finished quadratic equations can be solved through factoring. If the left-hand side of the general form of a quadratic equation can be factored, There is only one way for the factored equation to be true is for one or both of the factors to be zero. For instance, the left-hand side of the quadratic equation x2 + x - 6 = 0 can be factored within (x + 3)(x - 2). The only way for the equation (x + 3) (x - 2) = 0 to be true is for either (x + 3) or (x - 2) to be zero. Therefore, the roots of quadratic equations that can be factored can be found through setting each of the factors equal to zero and solving the resulting linear equations. Therefore, the roots of (x + 3)(x - 2) = 0 are found through 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 in which it is the reverse of multiplication. For instance, if we have two expressions (dx + c) and (cx + g) and multiply them, we acquire (using the distribution laws)
(dx + c) (fx + g) = (dx) (fx) + (dx) (g) + (c) (fx) + cg =
= dfx2 + (dg + cf)x + cg.
Therefore, a statement (dx + c) (fx + g) = 0 can be written
df x2 + (dg + cf)x + cg = 0.
Presently, if one is given an equation ax2 + bx + c = 0, he knows in which the symbol a is the product of two numbers (df) & 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, while they might be 1.5 and 2. The final -4 (c in the general equation) is the product of two numbers (eg), perhaps -2 and 2 or -1 and 4. These combinations are tried to see that provides the proper value of b (dg + ef), from above.