Consider the quadratic equation $4 x^2-11 x-3=0$. Which of the following shows this equation rewritten and ready to solve using factoring by grouping?
A) $4 x^2-6 x-5 x-3=0$
B) $4 x^2-12 x+x-3=0$
C) $4 x^2+12 x-x-3=0$
D) $4 x^2-9 x-2 x-3=0$
Answer (1)
The problem requires rewriting the quadratic equation 4 x 2 − 11 x − 3 = 0 for factoring by grouping.
We need to split the middle term − 11 x into two terms whose coefficients multiply to 4 × − 3 = − 12 .
Option B, 4 x 2 − 12 x + x − 3 = 0 , correctly splits the middle term since − 12 x + x = − 11 x and ( − 12 ) ( 1 ) = − 12 .
Therefore, the correct answer is 4 x 2 − 12 x + x − 3 = 0 .
Explanation
Understanding the Problem We are given the quadratic equation 4 x 2 − 11 x − 3 = 0 and asked to rewrite it in a form suitable for factoring by grouping. This means we need to split the middle term, − 11 x , into two terms such that the product of their coefficients equals the product of the leading coefficient (4) and the constant term (-3), which is 4 × − 3 = − 12 .
Checking the Options We need to find two numbers that add up to -11 and multiply to -12. Let's examine the options:
Option A: 4 x 2 − 6 x − 5 x − 3 = 0 . Here, − 6 x − 5 x = − 11 x , but ( − 6 ) × ( − 5 ) = 30 , which is not -12. So, this option is incorrect.
Option B: 4 x 2 − 12 x + x − 3 = 0 . Here, − 12 x + x = − 11 x , and ( − 12 ) × ( 1 ) = − 12 . This option satisfies both conditions.
Option C: 4 x 2 + 12 x − x − 3 = 0 . Here, 12 x − x = 11 x , which is not − 11 x . So, this option is incorrect.
Option D: 4 x 2 − 9 x − 2 x − 3 = 0 . Here, − 9 x − 2 x = − 11 x , but ( − 9 ) × ( − 2 ) = 18 , which is not -12. So, this option is incorrect.
Selecting the Correct Option Only option B satisfies both conditions: the middle term sums to − 11 x , and the product of the coefficients of the split middle terms is -12. Therefore, the correct rewritten equation is 4 x 2 − 12 x + x − 3 = 0 .
Examples
Factoring by grouping is a useful technique in many real-world applications. For example, consider a rectangular garden whose area is given by the quadratic expression 4 x 2 − 11 x − 3 . If you want to determine the dimensions of the garden in terms of x , you would need to factor this quadratic expression. Factoring by grouping helps break down the problem into smaller, manageable steps, allowing you to find the length and width of the garden. This technique is also used in engineering to optimize designs and in economics to model and solve optimization problems.
