factor 8m^2 - 71m - 9 using the quadratic formula

To factor the quadratic expression \( 8m^2 - 71m - 9 \) using the quadratic formula, follow these steps:

1. **Identify the coefficients**: The quadratic formula is given by:

   \[
   m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]

   For the quadratic equation \( 8m^2 - 71m - 9 = 0 \), the coefficients are:
   \[
   a = 8, \quad b = -71, \quad c = -9
   \]

2. **Calculate the discriminant** \( \Delta \):
   \[
   \Delta = b^2 - 4ac
   \]
   Substituting the values:
   \[
   \Delta = (-71)^2 - 4(8)(-9) = 5041 + 288 = 5329
   \]

3. **Solve for \( m \)** using the quadratic formula:
   \[
   m = \frac{-(-71) \pm \sqrt{5329}}{2(8)} = \frac{71 \pm 73}{16}
   \]
   
   This gives two possible solutions:
   \[
   m_1 = \frac{71 + 73}{16} = \frac{144}{16} = 9
   \]
   \[
   m_2 = \frac{71 - 73}{16} = \frac{-2}{16} = -\frac{1}{8}
   \]

4. **Write the factorized form**: Using the solutions \( m_1 = 9 \) and \( m_2 = -\frac{1}{8} \), the quadratic can be factored as:
   \[
   8m^2 - 71m - 9 = 8(m - 9)\left(m + \frac{1}{8}\right)
   \]
   Simplifying further:
   \[
   8m^2 - 71m - 9 = (8m + 1)(m - 9)
   \]

Thus, the factored form of \( 8m^2 - 71m - 9 \) is:
\[
(8m + 1)(m - 9)
\]

Would you like any further details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you verify that the factorization is correct?
2. What is the purpose of the discriminant in the quadratic formula?
3. Can every quadratic expression be factored using the quadratic formula?
4. How would the factorization change if the discriminant was negative?
5. How does the quadratic formula relate to completing the square?

**Tip**: Always check your factorization by expanding the factored form to ensure it matches the original quadratic expression.

Factor 8m^2 - 71m - 9 using the quadratic formula

Learn how to factor the quadratic expression 8m^2 - 71m - 9 using the quadratic formula. This problem covers key algebraic concepts, including factoring quadratic equations, using the quadratic formula, and calculating the discriminant. Suitable for students in Grades 9-11.

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