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Order the steps from first to last that would be used to factor:

3x2−13x−10


(7 points)
Arrange responses in the correct order to answer the question. Select a response, navigate to the desired position and insert response at that position. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can be moved using the up and down arrow keys or by dragging with a mouse.

1 answer

To factor the quadratic expression \( 3x^2 - 13x - 10 \), here are the steps arranged from first to last:

  1. Identify the quadratic: Recognize the expression \( 3x^2 - 13x - 10 \).
  2. Multiply the leading coefficient and the constant term: Multiply \( 3 \) (the coefficient of \( x^2 \)) and \( -10 \) (the constant term) to get \( -30 \).
  3. Find two numbers that multiply to -30 and add to -13: These two numbers are \( -15 \) and \( 2 \) since \( -15 + 2 = -13 \) and \( -15 \times 2 = -30 \).
  4. Rewrite the middle term: Rewrite \( -13x \) as \( -15x + 2x \).
  5. Group the terms: Reorganize the expression as \( (3x^2 - 15x) + (2x - 10) \).
  6. Factor by grouping: Factor out the common factors in each group:
    • From \( 3x^2 - 15x \), factor out \( 3x \): \( 3x(x - 5) \).
    • From \( 2x - 10 \), factor out \( 2 \): \( 2(x - 5) \).
  7. Combine the factors: Now combine the factored groups: \( 3x(x - 5) + 2(x - 5) \).
  8. Factor out the common binomial: Factor out \( (x - 5) \): \( (x - 5)(3x + 2) \).
  9. Final result: The factored form of the quadratic is \( (x - 5)(3x + 2) \).

So the ordered steps to factor the expression are:

  1. Identify the quadratic.
  2. Multiply the leading coefficient and the constant term.
  3. Find two numbers that multiply to -30 and add to -13.
  4. Rewrite the middle term.
  5. Group the terms.
  6. Factor by grouping.
  7. Combine the factors.
  8. Factor out the common binomial.
  9. Final result.