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Factor the trinomial completely.

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

2x^2 + 5x - 3

9x^2 - 9x - 4

3 answers

To factor the given trinomials, we will find two numbers that multiply to give the product of the leading coefficient and the constant term and add to give the middle coefficient.

  1. Factoring \(2x^2 + 5x - 3\):

    The product of the leading coefficient (2) and the constant term (-3) is: \[ 2 \times (-3) = -6 \] We need two numbers that multiply to \(-6\) and add to \(5\). The pair that works is \(-1\) and \(6\) (since \(-1 + 6 = 5\)).

    Rewrite the middle term: \[ 2x^2 - 1x + 6x - 3 \] Now, group: \[ (2x^2 - 1x) + (6x - 3) \] Factor out the common terms: \[ x(2x - 1) + 3(2x - 1) \] Now factor out the common binomial: \[ (2x - 1)(x + 3) \]

  2. Factoring \(9x^2 - 9x - 4\):

    The product of the leading coefficient (9) and the constant term (-4) is: \[ 9 \times (-4) = -36 \] We need two numbers that multiply to \(-36\) and add to \(-9\). The pair that works is \(-12\) and \(3\) (since \(-12 + 3 = -9\)).

    Rewrite the middle term: \[ 9x^2 - 12x + 3x - 4 \] Now, group: \[ (9x^2 - 12x) + (3x - 4) \] Factor out the common terms: \[ 3x(3x - 4) + 1(3x - 4) \] Now factor out the common binomial: \[ (3x - 4)(3x + 1) \]

Now putting the factored forms into the requested format:

| Expression | Factored Form | |-----------------------|-------------------------| | \(2x^2 + 5x - 3\) | \((2x - 1)(x + 3)\) | | \(9x^2 - 9x - 4\) | \((3x - 4)(3x + 1)\) |

that's not right

Let’s take another look at factoring both trinomials step by step.

1. Factoring \(2x^2 + 5x - 3\)

Step 1: Identify a, b, and c

  • Here, \( a = 2 \), \( b = 5 \), and \( c = -3 \).

Step 2: Calculate \( a \cdot c \):

  • \( a \cdot c = 2 \cdot (-3) = -6 \).

Step 3: Find two numbers that multiply to \(-6\) and add to \(5\):

  • The numbers are \(6\) and \(-1\) (since \(6 + (-1) = 5\)).

Step 4: Rewrite the middle term using these numbers: \[ 2x^2 + 6x - 1x - 3 \]

Step 5: Group the terms: \[ (2x^2 + 6x) + (-1x - 3) \]

Step 6: Factor each group: \[ 2x(x + 3) - 1(x + 3) \]

Step 7: Factor out the common binomial: \[ (2x - 1)(x + 3) \]

2. Factoring \(9x^2 - 9x - 4\)

Step 1: Identify a, b, and c

  • Here, \( a = 9 \), \( b = -9 \), and \( c = -4 \).

Step 2: Calculate \( a \cdot c \):

  • \( a \cdot c = 9 \cdot (-4) = -36 \).

Step 3: Find two numbers that multiply to \(-36\) and add to \(-9\):

  • The numbers are \(-12\) and \(3\) (since \(-12 + 3 = -9\)).

Step 4: Rewrite the middle term using these numbers: \[ 9x^2 - 12x + 3x - 4 \]

Step 5: Group the terms: \[ (9x^2 - 12x) + (3x - 4) \]

Step 6: Factor each group: \[ 3x(3x - 4) + 1(3x - 4) \]

Step 7: Factor out the common binomial: \[ (3x - 4)(3x + 1) \]

Final Factored Forms:

  • For \(2x^2 + 5x - 3\): \((2x - 1)(x + 3)\)
  • For \(9x^2 - 9x - 4\): \((3x - 4)(3x + 1)\)

Here they are in the requested format:

| Expression | Factored Form | |-----------------------|-------------------------| | \(2x^2 + 5x - 3\) | \((2x - 1)(x + 3)\) | | \(9x^2 - 9x - 4\) | \((3x - 4)(3x + 1)\) |

Thank you for your patience! If there’s anything else, feel free to ask.