Factoring with Grouping – Practice Problems

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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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Exercises in this Practice Problem
Establish the equation corresponding to the area of the façade of the building.
Solve the equation describing the area of the building to be painted.
Find the mistakes in Vincent's calculations.
Factor each trinomial.
Determine the solutions and decide which solution is a reasonable length.
Determine the solutions of the equation.