Factoring with Grouping – Practice Problems
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- Practice Problems
There are easy ways to solve this quadratic equations, like 3x2 - 25x + 56 = 14. One way is by combining factoring and grouping.
The standard form of the quadratic equation is given by ax2 + bx + c = 0.
Once such an equation is put into standard form, we can then determine its factors by finding the factors of a*c such that a+c=b.
For example, the equation 3x2 - 25x + 56 = 14 in standard form is 3x2 - 25x + 42 = 0, with a = 3, b = -25, and c = 42.
We can then see that (a * c) = 3 * 42 = 126, which has the factors:
Factors | 1,126 | -1,-126 | 2,63 | -2,-63 | 3,42 | -3,-42 | 6,21 | -6,-21 | 7,18 | -7,-18 |
Sum | 127 | -127 | 65 | -65 | 45 | -45 | 27 | -27 | 25 | -25 |
with (-7,-18) fulfilling the statement a + c = b: (-7) + (-18) = -25.
Rewriting our equation as 3x2 - 7x - 18x + 42 = 0, and grouping it as (3x2 - 7x) - (18x - 42) = 0,
we can factor x from the first group and 6 from the second group to get x (x - 7) - 6 (x - 7) = 0.
We can then isolate (x - 7) as a common factor and finally get the equation: (x - 6)(x - 7) = 0.
We have thus factored 3x2 - 25x + 42 = 0 into (x - 6)(x - 7) = 0, from which we can finally see that either x = 6 or x = 7.
Analyze Functions Using Different Representations.
CCSS.MATH.CONTENT.HSF.IF.C.8.A
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