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Type your answers to each problem in the table below. The order of the factors doesn't matter, but use parentheses around each one!

9x^2 - 9x - 4

5 answers

To factor the quadratic expression \(9x^2 - 9x - 4\), we will use the method of factoring by grouping or by applying the quadratic formula if necessary.

We can start by identifying the coefficients:

  • \(a = 9\)
  • \(b = -9\)
  • \(c = -4\)

The factored form of a quadratic \(ax^2 + bx + c\) can sometimes be found by looking for two numbers that multiply to \(ac\) (which is \(9 \times -4 = -36\)) and add to \(b\) (which is \(-9\)).

However, since simple factoring might be complex in this case, we can use the quadratic formula to find the roots first, which can help us express it in factored form:

The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Calculating the discriminant: \[ b^2 - 4ac = (-9)^2 - 4(9)(-4) = 81 + 144 = 225 \]

Now, applying the quadratic formula: \[ x = \frac{-(-9) \pm \sqrt{225}}{2(9)} = \frac{9 \pm 15}{18} \]

This results in two possible values for \(x\):

  1. \(x = \frac{24}{18} = \frac{4}{3}\)
  2. \(x = \frac{-6}{18} = -\frac{1}{3}\)

Now we can express the quadratic in factored form, using these roots: \[ 9x^2 - 9x - 4 = 9\left(x - \frac{4}{3}\right)\left(x + \frac{1}{3}\right) \]

To eliminate the fractions, multiply each factor by \(3\): \[ = 9\left(3x - 4\right)\left(3x + 1\right) \]

Thus, the factored form is: \[ (3x - 4)(3x + 1) \]

So, the answer in the requested format is: \[ (3x - 4)(3x + 1) \]

wrong

Apologies for any confusion in my previous response. Let's re-factor the expression \(9x^2 - 9x - 4\) step by step.

Step 1: Identify the Coefficients

  • \(a = 9\)
  • \(b = -9\)
  • \(c = -4\)

Step 2: Multiply \(a\) and \(c\)

Calculate \(ac = 9 \times -4 = -36\).

Step 3: Find Two Numbers

We need to find two numbers that multiply to \(-36\) and add to \(-9\). The suitable pair is:

  • \(3\) and \(-12\)

Step 4: Rewrite the Middle Term

We can rewrite the expression \(9x^2 - 9x - 4\) as follows: \[ 9x^2 + 3x - 12x - 4 \]

Step 5: Factor by Grouping

Now, group the terms: \[ (9x^2 + 3x) + (-12x - 4) \]

Factor out the common terms: \[ 3x(3x + 1) - 4(3x + 1) \]

Step 6: Factor Out the Common Binomial

Now factor out the common factor \((3x + 1)\): \[ (3x + 1)(3x - 4) \]

Final Answer

Thus, the factored form of \(9x^2 - 9x - 4\) is: \[ (3x + 1)(3x - 4) \]

Thank you for your patience, and I appreciate your understanding!

okay but its still wrong

I apologize for the confusion. Let's carefully go through factoring \(9x^2 - 9x - 4\) correctly this time.

Original Expression

We start with the quadratic expression: \[ 9x^2 - 9x - 4 \]

Step 1: Identify Coefficients

  • \(a = 9\)
  • \(b = -9\)
  • \(c = -4\)

Step 2: Use the Quadratic Formula

To find the roots, we use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Calculating the discriminant: \[ b^2 - 4ac = (-9)^2 - 4(9)(-4) = 81 + 144 = 225 \]

Step 3: Find Roots

Now apply the quadratic formula: \[ x = \frac{9 \pm \sqrt{225}}{2 \cdot 9} = \frac{9 \pm 15}{18} \]

Calculating both roots:

  1. \(x = \frac{24}{18} = \frac{4}{3}\)
  2. \(x = \frac{-6}{18} = -\frac{1}{3}\)

Step 4: Write in Factored Form

Using the roots, the expression can be written in its factored form: \[ 9\left(x - \frac{4}{3}\right)\left(x + \frac{1}{3}\right) \]

To eliminate the fractions, we can multiply each factor by 3: \[ = 9\left(3x - 4\right)\left(3x + 1\right) \]

Final Factored Form

Thus, the correct factored form of \(9x^2 - 9x - 4\) is: \[ (3x - 4)(3x + 1) \]

However, if we multiply it back out: \[ (3x - 4)(3x + 1) = 9x^2 + 3x - 12x - 4 = 9x^2 - 9x - 4 \] This confirms that the factorization is indeed correct.

If there is a specific form or method you are looking for, please let me know!