Welcome to factoring trinomials using the AC method. The AC method is a systematic technique for factoring trinomials of the form ax squared plus bx plus c, where a is not equal to zero. This method is particularly useful when the leading coefficient a is not equal to one. Let's explore this powerful factoring technique with a concrete example.
Let's start with step one of the AC method: identifying the coefficients. In our example trinomial 6x squared plus 7x minus 3, we identify a equals 6, which is the coefficient of x squared, b equals 7, which is the coefficient of x, and c equals negative 3, which is the constant term. These three values are crucial for the AC method.
Now for steps two and three. Step two: we calculate ac, which is 6 times negative 3, equals negative 18. Step three: we need to find two numbers p and q such that p times q equals negative 18, and p plus q equals 7. Let's check the factor pairs of negative 18. We have 1 and negative 18, but their sum is negative 17. Next, 2 and negative 9, but their sum is negative 7. Then 3 and negative 6, but their sum is negative 3. Finally, 9 and negative 2: their product is negative 18 and their sum is 7. Perfect! So p equals 9 and q equals negative 2.
Now for steps four and five. Step four: we rewrite the middle term 7x as 9x minus 2x, using our values p equals 9 and q equals negative 2. So 6x squared plus 7x minus 3 becomes 6x squared plus 9x minus 2x minus 3. Step five: we group the terms. The first group is 6x squared plus 9x, and the second group is negative 2x minus 3. From the first group, we factor out 3x to get 3x times 2x plus 3. From the second group, we factor out negative 1 to get negative 1 times 2x plus 3.
Finally, we complete the AC method by factoring out the common binomial. We have 3x times 2x plus 3, minus 1 times 2x plus 3. Notice that 2x plus 3 appears in both terms. We can factor this out to get 3x minus 1, times 2x plus 3. Let's verify our answer by expanding: 3x minus 1 times 2x plus 3 equals 6x squared plus 9x minus 2x minus 3, which simplifies to 6x squared plus 7x minus 3, our original trinomial. The AC method successfully factored our trinomial into two binomials!