In this lesson, we'll learn how to factor the quadratic expression x squared minus 7x plus 12. Factoring means finding two binomials that multiply together to give us our original expression. We're looking for something in the form x plus a times x plus b. When we expand this, we get x squared plus a plus b times x plus a times b. So we need to find values for a and b where a plus b equals negative 7 and a times b equals positive 12.
To factor our quadratic, we need to find two numbers that multiply to give 12 and add to give negative 7. Let's systematically list all factor pairs of 12. We have: 1 and 12, 2 and 6, 3 and 4, and their negative counterparts. Now let's calculate the sum for each pair. One plus twelve is thirteen. Two plus six is eight. Three plus four is seven. For the negative pairs: negative one plus negative twelve is negative thirteen. Negative two plus negative six is negative eight. And finally, negative three plus negative four equals negative seven. This is exactly what we need! Also, negative three times negative four equals positive twelve. So our factors are negative three and negative four.
Now that we've identified our factors as negative three and negative four, we can write the factored form of our quadratic. We substitute these values into our general form x plus a times x plus b. This gives us x plus negative three times x plus negative four. When we have addition of a negative number, we can rewrite it as subtraction. So x plus negative three becomes x minus three, and x plus negative four becomes x minus four. Therefore, our factored form is x minus three times x minus four. Let's verify this is correct by expanding it.
To verify our factorization is correct, let's expand x minus three times x minus four using the FOIL method. FOIL stands for First, Outer, Inner, Last - referring to which terms we multiply together. First: x times x equals x squared. Outer: x times negative four equals negative four x. Inner: negative three times x equals negative three x. Last: negative three times negative four equals positive twelve. Now we combine all these terms: x squared minus four x minus three x plus twelve. Finally, we combine like terms: negative four x minus three x equals negative seven x. So our expanded form is x squared minus seven x plus twelve, which matches our original expression. This confirms our factorization is correct.
There are alternative methods we can use to factor quadratics. Let's briefly look at two: the AC method and completing the square. The AC method starts by identifying coefficients a, b, and c in our standard form ax squared plus bx plus c. Here, a equals one, b equals negative seven, and c equals twelve. We then calculate a times c, which is one times twelve equals twelve. Next, we find factors of twelve that sum to our b value of negative seven. As we found before, negative three and negative four work because they sum to negative seven and multiply to twelve. This leads us to the same factored form: x minus three times x minus four. Completing the square is another approach where we manipulate the equation to form a perfect square trinomial. While more complex for this example, it's a powerful technique for solving quadratic equations and also leads to the same result.
Let's practice with another quadratic: x squared minus five x plus six. We need to find two numbers that multiply to six and add to negative five. Let's list the factor pairs of six: one and six, two and three, and their negative counterparts. Calculating the sums: one plus six is seven, two plus three is five, negative one plus negative six is negative seven, and negative two plus negative three is negative five. This is what we need! Also, negative two times negative three equals six. So our factors are negative two and negative three, giving us the factored form x minus two times x minus three. Let's verify: x minus two times x minus three equals x squared minus three x minus two x plus six, which simplifies to x squared minus five x plus six. This matches our original expression, confirming our answer is correct.