Factor to find the zeros of the function defined by the quadratic expression.
12x2 − 156x + 480
To factor the quadratic expression 12x^2 - 156x + 480, we'll look for two binomials in the form (ax + b)(cx + d) that multiply to give the original expression.
First, we'll find the product of a and c: ac = 12 * 480 = 5760.
Now we need to find two numbers whose product is 5760 and whose sum is -156. By factoring 5760, we find that 5760 = 6 * 960 = 12 * 480 = 15 * 384 = 16 * 360 = 20 * 288 = 24 * 240 = 30 * 192 = 32 * 180 = 40 * 144 = 48 * 120.
From these pairs, we see that 48 and 120 have a sum of -156. Therefore, we can use these numbers as the coefficients b and d.
So our expression can be factored as:
12x^2 - 156x + 480 = (ax + 48)(cx + 120).
To find the values of a and c, we can find the coefficients of the other terms by multiplying out the expression:
(ax + 48)(cx + 120) = acx^2 + (120a + 48c)x + 120 * 48.
We know that a * c = 12, so we need to find two numbers whose product is 12 and whose sum is 120. These numbers are 6 and 2.
Therefore, we can factor the expression as:
12x^2 - 156x + 480 = (6x + 48)(2x + 120).
The zeros of the function defined by this quadratic expression occur when the expression equals zero. So, setting each factor equal to zero gives the equations:
6x + 48 = 0 and 2x + 120 = 0.
Solving these equations gives the zeros:
6x = -48 -> x = -8
2x = -120 -> x = -60.
So the zeros of the function are -8 and -60.
To find the zeros of the quadratic expression 12x^2 - 156x + 480, we need to factor it.
Step 1: Find the common factor, if any.
The common factor in this expression is 12, so we factor it out:
12(x^2 - 13x + 40)
Step 2: Factor the quadratic expression inside the parentheses.
We need to find two numbers that multiply to give 40 and add up to -13.
The numbers are -8 and -5, because (-8) * (-5) = 40, and (-8) + (-5) = -13.
So, we can factor the quadratic expression as:
12(x - 8)(x - 5)
Now, we have factored the quadratic expression completely.
To find the zeros, we set the expression equal to zero and solve for x:
12(x - 8)(x - 5) = 0
Setting each factor equal to zero:
x - 8 = 0 --> x = 8
x - 5 = 0 --> x = 5
Therefore, the zeros of the quadratic function defined by the expression 12x^2 - 156x + 480 are x = 8 and x = 5.