Question

Part A: Show all work to solve the quadratic equation x2 − 12x + 35 = 0 by factoring. (2 points)

Part B: Using complete sentences, explain what the solutions from Part A represent on the graph. (2 points)

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Answer 1

A) x = {5, 7}

B) The solutions make the equation true.

Explanation:

Part A:

To solve this by factoring, you need to find factors of 35 that have a sum of -12. Since 35 is the product of two primes, the search is a short one.

35 = (-1)(-35) = (-5)(-7)

The corresponding sums are -36 and -12, so the latter factor pair is the one we want. Since the coefficient of x^2 is 1, we can use these numbers directly in the binomial factors:

x^2 -12x +35 = (x -5)(x -7) = 0

The zero product rule tells us this product is zero only when one of the factors is zero:

x -5 = 0 ⇒ x = 5

x -7 = 0 ⇒ x = 7

The two solutions are x=5 and x=7.

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Part B:

The solutions from part A are the x-intercepts of the graph of the quadratic expression. They are the values of x that make the quadratic expression be zero. That is, they are the values of x that make the equation true.