To factor the trinomial \(9x^2 - 9x - 4\), we can use the method of factoring by grouping or applying the quadratic formula to find the roots first.
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Identify \( a = 9 \), \( b = -9 \), and \( c = -4 \).
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Calculate the discriminant: \[ D = b^2 - 4ac = (-9)^2 - 4(9)(-4) = 81 + 144 = 225 \]
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The square root of the discriminant: \[ \sqrt{D} = \sqrt{225} = 15 \]
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Now use the quadratic formula to find the roots: \[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-(-9) \pm 15}{2(9)} = \frac{9 \pm 15}{18} \]
- For \( x_1 = \frac{24}{18} = \frac{4}{3} \)
- For \( x_2 = \frac{-6}{18} = -\frac{1}{3} \)
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Now we can write the trinomial in its factored form based on its roots: \[ 9x^2 - 9x - 4 = 9\left(x - \frac{4}{3}\right)\left(x + \frac{1}{3}\right) \]
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To express it without fractions, multiply the factors by 3: \[ = (3x - 4)(3x + 1) \]
So, the factored form is: \[ (3x - 4)(3x + 1) \]
Here is how you can summarize it in the table format you mentioned:
| Factor 1 | Factor 2 | |----------|------------| | (3x - 4) | (3x + 1) |