Verified Answer :

  • Identify coefficients \(a=11, b=9, c=-2\).
  • Find two numbers whose product is \(ac = -22\) and whose sum is \(b = 9\), which are \(11\) and \(-2\).
  • Split the middle term: \(11v^2 + 11v - 2v - 2\).
  • Factor by grouping to obtain the final result: \(\boxed{(11v - 2)(v + 1)}\).

Explanation

  1. Analysis of the Quadratic Expression

To factor the quadratic expression \(11v^2 + 9v - 2\), we look for a representation in the form of two binomials. This is a quadratic expression with coefficients \(a = 11\), \(b = 9\), and \(c = -2\). The most systematic way to factor this is by using the AC method, where we aim to split the middle term into two parts such that we can factor the expression by grouping.

  1. Finding the Factors for Splitting the Middle Term

We need to find two numbers that multiply to the product of \(a\) and \(c\), and add up to \(b\).
First, we calculate the product of \(a\) and \(c\):
\[a \times c = 11 \times (-2) = -22.\]
Next, we seek two numbers \(m\) and \(n\) such that:
\[m \times n = -22\]
\[m + n = 9\]
Testing pairs of factors of -22:

  • \(1 \times (-22) = -22\), \(1 + (-22) = -21\)
  • \(-1 \times 22 = -22\), \(-1 + 22 = 21\)
  • \(2 \times (-11) = -22\), \(2 + (-11) = -9\)
  • \(-2 \times 11 = -22\), \(-2 + 11 = 9\)

The pair \(-2\) and \(11\) satisfies both conditions.

  1. Factoring by Grouping and Final Result

Now, we rewrite the original middle term \(9v\) using the integers -2 and 11 found in the previous step:
\[11v^2 + 9v - 2 = 11v^2 + 11v - 2v - 2.\]
Next, we perform factoring by grouping. We group the first two terms and the last two terms:
\[(11v^2 + 11v) + (-2v - 2).\]
Now, we factor out the greatest common factor from each group:

  • From the first group \((11v^2 + 11v)\), we can factor out \(11v\): \(11v(v + 1)\).
  • From the second group \((-2v - 2)\), we can factor out \(-2\): \(-2(v + 1)\).

Putting them together, we have:
\[11v(v + 1) - 2(v + 1).\]
Finally, we can extract the common binomial factor \((v + 1)\):
\[(11v - 2)(v + 1).\]
Thus, the expression is completely factored as \((11v - 2)(v + 1)\).

Examples

Factoring quadratics is a foundational skill in algebra that allows us to find the zeros of a function, which is essential for solving real-world problems. For example, if you are modeling the trajectory of a projectile, the quadratic equation describes its height at a given time \(v\). By factoring the expression \(11v^2 + 9v - 2 = 0\), you can identify the specific times when the projectile hits the ground or reaches certain heights, allowing you to solve complex motion problems with ease.

Related Questions