Use the zero product property to find the solution to the equation 6x^2 -5x =56
Accepted Solution
A:
Answer: x = {-8/3, 7/2}Step-by-step explanation:To use the zero product property, you need the equation in the form of a product that is equal to zero. We can get that by subtracting 56 and factoring the resulting equation. 6x^2 -5x -56 = 0 (3x +8)(2x -7) = 0The zero product property tells us this product is zero only when the factors are zero: 3x +8 = 0 ⇒ x = -8/3 2x -7 = 0 ⇒ x = 7/2The solution is x = {-8/3, 7/2}._____Comment on factoringConsider the product ... (ax +b)(cx +d) = (ac)x^2 +(ad +bc)x +(bd)Now consider the product of first and last term coefficients compared to the coefficient of the middle term: acbd = (ad)(bc) vs. ad+bcWe see that the coefficient of the middle term is the sum of two of the factors of the first·last product. This means we want factors of 6·(-56) that have a sum of -5. 6(-56) = -(2^4)(3)(7) . . . . has 20 divisors. We're looking for factors that are nearly equal. The clue is given by simplifying the above factoring: = -(16)(21) = (16)(-21) . . . . the sum of these factors is -5, as we need.Again considering the first-last product and the middle coefficient, we see that we can choose ... ad = -21, bc = 16; ac = 6, so a=3, c=2, and ... (a, b, c, d) = (3, 8, 2, -7)These values give us the factors we used above.Note this process gets easier with practice and familiarity with multiplication tables.