6.4 Quadratic equations  (Page 2/14)

To factor $x^2+x-6=0,$ we look for two numbers whose product equals $\mathrm{-6}$ and whose sum equals 1. Begin by looking at the possible factors of $\mathrm{-6.}$

The last pair, $3\cdot \left(\mathrm{-2}\right)$ sums to 1, so these are the numbers. 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.

The two solutions are $2$ and $\mathrm{-3.}$ We can see how the solutions relate to the graph in [link] . The solutions are the x- intercepts of $x^2+x-6=0.$

Find two numbers whose product equals $15$ and whose sum equals $8.$ List the factors of $15.$

The numbers that add to 8 are 3 and 5. Then, write the factors, set each factor equal to zero, and solve.

The solutions are $\mathrm{-3}$ and $\mathrm{-5.}$

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.

The solutions are $3$ and $\mathrm{-3.}$