Answers:
A = 3
B = -7
C = 3
D = -7
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Explanation:
You could use the AC method to factor, but I think the method described below is a more efficient way of factoring.
Set the given expression equal to zero and solve for x using the quadratic formula. So the goal is to find the roots for [tex]9x^2-42x+49=0[/tex]
We'll plug in a = 9, b = -42, and c = 49.
[tex]x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x = \frac{-(-42)\pm\sqrt{(-42)^2-4(9)(49)}}{2(9)}\\\\x = \frac{42\pm\sqrt{0}}{18}\\\\x = \frac{42+0}{18}\\\\x = \frac{42}{18}\\\\x = \frac{7}{3}\\\\[/tex]
Because the root is x = 7/3, it leads to the factor 3x-7 = 0
Note that we go from x = 7/3 to 3x = 7 after multiplying both sides by 3. Then we subtract 7 from both sides.
It turns out that we have two copies of the identical factor (3x-7) because x = 7/3 is a double root.
So in reality we have the factorization (3x-7)(3x-7) which we can shorten into (3x-7)^2 if we wanted.
Use the box method or FOIL method to expand (3x-7)(3x-7) and you should get 9x^2-21x-21x+49 which simplifies to 9x^2-42x+49. This confirms we have the correct factorization.
Therefore, A and C are both 3, while B and D are both -7.
