So, in that sense, the operation of multiplication undoes the operation of factoring. The product is a quadratic expression. If we were to factor the equation, we would get back the factors we multiplied. The process of factoring a quadratic equation depends on the leading coefficient, whether it is 1 or another integer.
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.
A quadratic equation is an equation containing a second-degree polynomial; for example. We have one method of factoring quadratic equations in this form. Note that only one pair of numbers will work. Then, write the factors. To solve this equation, we use the zero-product property. Set each factor equal to zero and solve. We can see how the solutions relate to the graph in Figure 2. The numbers that add to 8 are 3 and 5.
Then, write the factors, set each factor equal to zero, and solve. Recognizing that the equation represents the difference of squares, we can write the two factors by taking the square root of each term, using a minus sign as the operator in one factor and a plus sign as the operator in the other.
Solve using the zero-factor property. When the leading coefficient is not 1, we factor a quadratic equation using the method called grouping, which requires four terms. Factor the first two terms, and then factor the last two terms.
This equation does not look like a quadratic, as the highest power is 3, not 2. Recall that the first thing we want to do when solving any equation is to factor out the GCF, if one exists. And it does here. The area of a rectangular garden is 30 square feet. If the length is 7 feet longer than the width, find the dimensions. Subtract 30 from both sides to set the equation equal to 0.
Use the Zero Product Property to solve for w. So, the width is 3 feet. The width of the garden is 3 feet, and the length is 10 feet. The example below shows another quadratic equation where neither side is originally equal to zero.
Note that the factoring sequence has been shortened. To make this side equal to 0, add 12 b to both sides. Factor out 5 b from the first pair and 2 from the second pair. Apply the Zero Product Property. If you factor out a constant, the constant will never equal 0.
So it can essentially be ignored when solving. See the following example. A small toy rocket is launched from a 4-foot pedestal. How long will it take the rocket to hit the ground? The rocket will be on the ground when the height is 0. So, substitute 0 for h in the formula. Factor the trinomial by grouping. Use the Zero Product Property. There is no need to set the constant factor -1 to zero, because -1 will never equal zero.
Interpret the answer. The rocket will hit the ground 4 seconds after being launched. A Correct. The original equation has 48 on the right. Then factor out the common factor, Then set the trinomial to 0 and solve for m.
You find that. You probably either factored the quadratic incorrectly or you solved the individual equations incorrectly. To use the Zero Product Property, one side must be 0.
You can find the solutions, or roots, of quadratic equations by setting one side equal to zero, factoring the polynomial, and then applying the Zero Product Property. Once the polynomial is factored, set each factor equal to zero and solve them separately. The answers will be the set of solutions for the original equation.
Not all solutions are appropriate for some applications. In many real-world situations, negative solutions are not appropriate and must be discarded. Example Problem The area of a rectangular garden is 30 square feet. Answer The width of the garden is 3 feet, and the length is 10 feet. Example Problem A small toy rocket is launched from a 4-foot pedestal.
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