factor the following expression into a product of two binomials.
64x^2-64x+16
Find two factors that multiply to the end terms and add to the value of the center term. Since the center term is negative and the right term is positive, both of the second term factors must be negative.
(8x-4)^2
To factor the expression 64x^2 - 64x + 16 into a product of two binomials, follow these steps:
Step 1: Look for a common factor, if any.
In this case, 64 is a common factor of all three terms: 64x^2, -64x, and 16. Divide each term by 64 to simplify the expression:
64x^2 / 64 = x^2
-64x / 64 = -x
16 / 64 = 1/4
Step 2: Write the simplified expression:
(x^2 - x + 1/4)
Step 3: Identify the pattern for squaring a binomial - (a-b)^2 = a^2 - 2ab + b^2.
In this case, (a-b)^2 = x^2 - 2x(1/2) + (1/2)^2
Step 4: Simplify the second term.
-2x(1/2) = -x
Step 5: Write the simplified expression using the squared binomial pattern:
(x - 1/2)^2
Therefore, the factored form of the expression 64x^2 - 64x + 16 is (x - 1/2)^2.