Factoring Quadratic Equations

Factoring Quadratic Equations Discover the Factoring Quadratic Equation’s Little-Known Secret of Math! If you’re looking for a fast, easy, and legitimate way to factoring quadratic equations, listen to this:.   A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable. Linear equations can have one or more variables. Linear equations occur with great regularity in applied math. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some “background” state in math, such as this factoring quadratic equations.

Factoring Quadratic Equations   Certain complete quadratic equations can be solved by factoring. If the left-hand side of the general form of a quadratic equation can be factored, the only way for the factored equation to be true is for one or both of the factors to be zero. For example, the left-hand side of the quadratic equation can be factored into (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. Thus, the roots of quadratic equations which can be factored can be found by setting each of the factors equal to zero and solving the resulting linear equations. Thus, the roots of (x + 3)(x – 2) = 0 are found by 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. For example, if we have two expressions (dx + c) and (cx + g) and multiply them, we obtain (using the distribution laws) Thus, a statement (dx + c) (fx + g) = 0 can be written Now, if one is given an equation , he knows that the symbol a is the product of two numbers (df) and c is also the product of two numbers. For the example , it is a reasonable guess that the numbers multiplying 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 two numbers (eg), perhaps -2 and 2 or -1 and 4. These combinations are tried to see which gives the proper 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 the general quadratic form 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.   Quadratic equations in which the numerical constant c is zero can always be solved by factoring. One of the two roots is zero. For example, the quadratic equation can be solved by factoring. The factors are (x) and (2x + 3). Thus, the roots are . If a quadratic equation in which the numerical constant c is zero is written in general form, a general expression can be written for its roots. The general form of a quadratic equation in which the numerical constant c is zero is the following: The left-hand side of this equation can be factored by removing an x from each term. x(ax + b) = 0 The roots of this quadratic equation are found by setting the two factors equal to zero and solving the resulting equations. Thus, the roots of a quadratic equation in which the numerical constant c is zero are x = 0 and   Example: Find the roots of the following quadratic equation. Solution: Using formulae above, one root is determined. x = 0 Using formulae above, substitute the values of a and b and solve for x.

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