Answer:
x = {-8/3, 7/2}
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) = 0
The 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 = 7/2
The solution is x = {-8/3, 7/2}.
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Comment on factoring
Consider 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+bc
We 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.