Write the quadratic equation in factored form. Be sure to write the entire equation.
x^2 - 5x - 24 = 0
Ask yourself what pair of numbers has a product of 24 and a difference of 5? That would be 8 and 3. Adjust the signs as needed.
(x-8)(x+3) = 0
To write the quadratic equation in factored form, we need to express it as a product of two binomials. Follow these steps to factor the given quadratic equation:
Step 1: Set up the equation
The given quadratic equation is x^2 - 5x - 24 = 0.
Step 2: Find two numbers whose product is equal to the constant term (in this case, -24) and whose sum is equal to the coefficient of the linear term (in this case, -5).
For -24, the possible pairs of numbers that multiply to give -24 are:
(1, -24), (2, -12), (3, -8), (4, -6), (-1, 24), (-2, 12), (-3, 8), (-4, 6).
Among these pairs, we need to find the pair whose sum is -5, which is (-8, 3).
Step 3: Rewrite the middle term of the quadratic equation using the two numbers found in Step 2.
The middle term of the quadratic equation is -5x. We rewrite it as -8x + 3x.
Step 4: Group the terms and factor by grouping.
Rearrange the quadratic equation by grouping the terms as follows:
(x^2 - 8x) + (-3x - 24) = 0
Now, factor out the Greatest Common Factor (GCF) from each group:
x(x - 8) - 3(x + 8) = 0
Step 5: Factor again.
By factoring out the GCF, we get:
x(x - 8) - 3(x + 8) = 0
x(x - 8) - 3(x + 8) = 0
Step 6: Rewrite the factored equation.
The quadratic equation in factored form is:
(x - 8)(x + 3) = 0
Thus, the entire equation in factored form is:
(x - 8)(x + 3) = 0