We are given the quadratic expression x squared minus 8x plus 15. Our goal is to factor this expression into two binomials. When we factor a quadratic of the form x squared plus bx plus c, we look for two numbers that multiply to c and add to b. In this case, we need two numbers that multiply to 15 and add to negative 8.
Let's identify the components of our quadratic expression. The coefficient of x squared is 1, the coefficient of x is negative 8, and the constant term is 15. We need to find two numbers that multiply to give 15 and add to give negative 8. This corresponds to the general pattern where (x plus a) times (x plus b) equals x squared plus (a plus b)x plus ab.
Now let's find all the factor pairs of 15. First we have 1 and 15, which multiply to 15 and add to 16. Next is 3 and 5, which multiply to 15 and add to 8. Then we have the negative pairs: negative 1 and negative 15 multiply to 15 but add to negative 16. Finally, negative 3 and negative 5 multiply to 15 and add to negative 8. This is exactly what we need! So negative 3 and negative 5 are our two numbers.
Now we can construct the factored form using our numbers negative 3 and negative 5. We write (x plus negative 3) times (x plus negative 5), which simplifies to (x minus 3) times (x minus 5). Let's verify this using the FOIL method. First: x times x equals x squared. Outer: x times negative 5 equals negative 5x. Inner: negative 3 times x equals negative 3x. Last: negative 3 times negative 5 equals 15. Combining like terms: x squared minus 5x minus 3x plus 15 equals x squared minus 8x plus 15. This matches our original expression!
Let's try an alternative method called the AC method. First, we identify our coefficients: a equals 1, b equals negative 8, and c equals 15. Then we calculate a times c, which is 1 times 15 equals 15. Now we need two numbers that multiply to 15 and add to negative 8. As we found before, these are negative 3 and negative 5. Next, we rewrite the middle term: x squared minus 3x minus 5x plus 15. Now we factor by grouping. Group the first two terms and the last two terms. Factor out x from the first group to get x times (x minus 3). Factor out negative 5 from the second group to get negative 5 times (x minus 3). Now we see that (x minus 3) is a common factor. Factoring it out gives us (x minus 5) times (x minus 3). This is the same result as our previous method!
Let's verify our factorization and explore its applications. We can verify by expanding (x minus 3) times (x minus 5) using the distributive property or an area model, both confirming we get x squared minus 8x plus 15. More importantly, factoring helps us find the zeros of the function. When (x minus 3) equals zero, x equals 3. When (x minus 5) equals zero, x equals 5. These are the x-intercepts of the parabola. On the graph of y equals x squared minus 8x plus 15, we can see the parabola crosses the x-axis at x equals 3 and x equals 5. The vertex of the parabola is at the point (4, negative 1).