Answered

Factor by grouping (sometimes called the ac-method).

$11 x^2-7 x-4$



First, choose a form with appropriate signs.

Then, fill in the blanks with numbers to be used for grouping.

Finally, show the factorization.



Form:

◯ $11 x^2+\square x+\square x-4$

◯ $11 x^2+\square x-\square x-4$

◯ $11 x^2-\square x+\square x-4$

◯ $11 x^2-\square x-\square x-4$



Factorization:

□

Verified Answer :

  • Identify coefficients \(a=11\), \(b=-7\), \(c=-4\) and calculate \(a \cdot c = -44\).
  • Find two numbers that multiply to \(-44\) and add to \(-7\), which are \(-11\) and \(4\).
  • Rewrite the expression as \(11x^2 - 11x + 4x - 4\) and group terms to factor out common expressions.
  • The final factored form is \(\boxed{(11x + 4)(x - 1)}\).

Explanation

  1. Analyzing the quadratic expression

To factor the quadratic expression \(11x^2 - 7x - 4\) using the ac-method, we first identify the coefficients \(a\), \(b\), and \(c\) from the general quadratic form \(ax^2 + bx + c\). Here, \(a = 11\), \(b = -7\), and \(c = -4\). The goal of the ac-method is to split the middle term \(bx\) into two terms \(px\) and \(qx\) such that their sum is \(b\) and their product is \(a \cdot c\).

  1. Finding the split terms

Calculate the product \(a \cdot c = 11 \cdot (-4) = -44\). We need to find two numbers that multiply to \(-44\) and add up to the middle coefficient \(b = -7\). Listing the pairs of factors of \(-44\), we find:
\(1 \cdot (-44) = -44, 1 + (-44) = -43\)
\(-1 \cdot 44 = -44, -1 + 44 = 43\)
\(2 \cdot (-22) = -44, 2 + (-22) = -20\)
\(-2 \cdot 22 = -44, -2 + 22 = 20\)
\(4 \cdot (-11) = -44, 4 + (-11) = -7\)
The numbers are \(4\) and \(-11\). These numbers will be used to split the middle term \(-7x\) into \(-11x + 4x\).

  1. Factorization by grouping

We can now write the expression as a four-term polynomial by replacing \(-7x\) with \(-11x + 4x\):
\[11x^2 - 11x + 4x - 4\]
This corresponds to the third form provided in the options:
\[11x^2 - 11x + 4x - 4\]
Grouping the terms, we have \((11x^2 - 11x) + (4x - 4)\). We factor out the greatest common factor from each group:
\(11x(x - 1) + 4(x - 1)\).
Finally, we factor out the common binomial factor \((x - 1)\):
\((11x + 4)(x - 1)\).

Examples

In engineering, understanding how to factor quadratic expressions is essential when solving for roots of equations that describe physical systems. For example, if the displacement of a structural component over time follows a quadratic path like \(11t^2 - 7t - 4 = 0\), finding the roots helps identify the exact moments when the displacement is zero, allowing for precise design and safety verification. Factoring by grouping is a standard, reliable technique that simplifies complex polynomial equations into manageable linear components.

Related Questions