HOW TO Given a quadratic equation with the leading coefficient of, factor it.ġ. We have one method of factoring quadratic equations in this form. In the quadratic equation, the leading coefficient, or the coefficient of, is. Solving Quadratics with a Leading Coefficient of 1 Where, and are real numbers, and if, it is in standard form. ![]() If then or ,where and are real numbers or algebraic expressions.Ī quadratic equation is an equation containing a second-degree polynomial for example THE ZERO-PRODUCT PROPERTY AND QUADRATIC EQUATIONS We can use the zero-product property to solve quadratic equations in which we first have to factor out the greatest common factor (GCF), and for equations that have special factoring formulas as well, such as the difference of squares, both of which we will see later in this section. We will look at both situations but first, we want to confirm that the equation is written in standard form,, where, and are real numbers, and. The process of factoring a quadratic equation depends on the leading coefficient, whether it is or another integer. If we were to factor the equation, we would get back the factors we multiplied. Set equal to zero, is a quadratic equation. For example, expand the factored expression by multiplying the two factors together. So, in that sense, the operation of multiplication undoes the operation of factoring. Multiplying the factors expands the equation to a string of terms separated by plus or minus signs. In other words, if the product of two numbers or two expressions equals zero, then one of the numbers or one of the expressions must equal zero because zero multiplied by anything equals zero. Solving by factoring depends on the zero-product property, which states that if, then or, where and are real numbers or algebraic expressions. If a quadratic equation can be factored, it is written as a product of linear terms. Factoring means finding expressions that can be multiplied together to give the expression on one side of the equation. Often the easiest method of solving a quadratic equation is factoring. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics. For example, equations such as and are quadratic equations. In these cases it is usually better to solve by completing the square or using the quadratic formula.An equation containing a second-degree polynomial is called a quadratic equation. However, not all quadratic equations can be factored evenly. (1,180) (2,90) (3,60) (4,45) (5,36) (6,30) ģ.2: p = -180, a negative number, therefore one factor will be positive and the other negative.ģ.3: b = 24, a positive number, therefore the larger factor will be positive and the smaller will be negative.įactoring quadratics is generally the easier method for solving quadratic equations. Is negative then one factor will be positive and the other negative. This equation is already in the proper form where a = 15, b = 24 and c = -12. Step 1: Write the equation in the general form ax 2 + bx + c = 0. This equation is already in the proper form where a = 4, b = -19 and c = 12.ģ.2: p = 48, a positive number, therefore both factors will be positive or both factors will be negative.ģ.3: b = -19, a negative number, therefore both factors will be negative. Step 8: Set each factor to zero and solve for x. ![]() Now that the equation has been factored, solve for x. Using the reverse of the distributive property we can write the outside expressions (shown in red in Step 6) as a second polynomial factor. If this does not occur, regroup the terms and try again. Notice that the parenthetical expression is the same for both groups. Step 7: Rewrite the equation as two polynomial factors. Step 6: Factor the greatest common denominator from each group. Step 4: Rewrite bx as a sum of two x -terms using the factor pair found in Step 3. If p is negative and b is positive, the larger factor will be positive and the smaller will be negative.ģ.2: p = 12, a positive number, therefore both factors will be positive or both factors will be negative.ģ.3: b = 7, a positive number, therefore both factors will be positive. If p is positive and b is negative, both factors will be negative. ![]() If both p and b are negative, the larger factor will be negative and the smaller will be positive. If both p and b are positive, both factors will be positive. If p is negative then one factor will be positive and the other negative.ģ.3: Determine the factor pair that will add to give b. If p is positive then both factors will be positive or both factors will be negative. Step 3: Determine the factor pairs of p that will add to b.įirst ask yourself what are the factors pairs of p, ignoring the negative sign for now.
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