Factor x^2+5x-24 completely.
In your mathematics course, your final grade was twelve points from a B. Your total points for the course were 252. How many points were possible for the course? (Assume 80% of the course total was a B.)
You factor
(x-3)(x+8)
x= 3,-8
To factor the quadratic expression x^2 + 5x - 24 completely, we need to find two binomial expressions whose product equals the given expression.
Step 1: Multiply the coefficient of x^2 and the constant term, ignoring the signs. In this case, it's (1)(-24) = -24.
Step 2: Find two numbers whose product is equal to -24 and whose sum is equal to the coefficient of the linear term, which is 5.
The numbers that satisfy these conditions are 8 and -3 because 8 * (-3) = -24 and 8 + (-3) = 5.
Step 3: Now, rewrite the linear term (5x) using the two numbers found in Step 2 (8 and -3). Replace 5x with 8x - 3x.
So, x^2 + 5x - 24 becomes x^2 + 8x - 3x - 24.
Step 4: Factor by grouping. Group the first two terms and the last two terms separately.
(x^2 + 8x) - (3x + 24).
Now, factor out the common terms from each group.
x(x + 8) - 3(x + 8).
Step 5: Notice that (x + 8) is a common binomial factor. Factor it out.
(x - 3)(x + 8).
Therefore, the factored form of the expression x^2 + 5x - 24 is (x - 3)(x + 8).