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Analyzing a Method of Factorization



Ginny factored $6 x^2-31 x-30$ as shown:

Step 1: $a c=-180$ and $b=31$

Step 2: $36(-5)=-180$ and $36+(-5)=31$

Step 3: $6 x^2+36 x-5 x-30$

Step 4: $6 x(x+6)-5(x+6)$

Step 5: $(x+6)(6 x-5)$



Determine if Ginny factored correctly. If not, explain where she made an error.


AGinny made a mistake in step 1 when she identified $b=31$. It should be $b=-31$.
BGinny was correct until step 3 when she used 36 and -5 as the coefficients of $x$.
CGinny was correct until step 4 when she incorrectly factored the GCF from $5 x-30$.
DGinny factored the trinomial correctly.

Verified Answer :

  • The quadratic expression is \(6x^2 - 31x - 30\), where the coefficient \(b\) is \(-31\).
  • Ginny incorrectly identified \(b = 31\) in step 1, leading her to choose factors \(36\) and \(-5\) that sum to \(31\) instead of \(-31\).
  • The correct factors that sum to \(-31\) and multiply to \(ac = -180\) are \(-36\) and \(5\).
  • The correct factorization is \(\boxed{\text{Ginny made a mistake in step 1 when she identified } b = 31. \text{ It should be } b = -31.}\)

Explanation

  1. Analyzing the Problem Data

We are tasked with verifying if the polynomial \(6x^2 - 31x - 30\) was factored correctly by Ginny. The standard quadratic trinomial form is \(ax^2 + bx + c\), where in our case \(a = 6\), \(b = -31\), and \(c = -30\). The 'ac-method' of factoring requires finding two numbers that multiply to the product of \(a\) and \(c\) (which is \(ac = 6 \times (-30) = -180\)) and add to the coefficient \(b = -31\).

  1. Identifying Ginny's Error

Ginny's work started with the claim \(ac = -180\) and \(b = 31\). Comparing this to our identified coefficients, we see that while \(ac\) was correctly calculated as \(-180\), the coefficient \(b\) was identified as \(31\) instead of \(-31\). This initial mistake cascaded through the subsequent steps, as Ginny proceeded to find factors that satisfy the conditions for \(b = 31\) instead of the actual \(b = -31\). Specifically, she chose \(36\) and \(-5\), which satisfy \(36 + (-5) = 31\) and \(36 \times (-5) = -180\). However, for our expression, we need two numbers that multiply to \(-180\) and add to \(-31\), which are \(-36\) and \(5\).

  1. Correcting the Factorization

By using \(36\) and \(-5\), Ginny formed the expression \(6x^2 + 36x - 5x - 30\), which simplifies to \(6x^2 + 31x - 30\). This is not equivalent to the original expression \(6x^2 - 31x - 30\). If we correctly identify the factors as \(-36\) and \(5\), we can rewrite the expression as \(6x^2 - 36x + 5x - 30\). Factoring by grouping gives \(6x(x-6) + 5(x-6) = (6x+5)(x-6)\). Expanding this result confirms: \((6x+5)(x-6) = 6x^2 - 36x + 5x - 30 = 6x^2 - 31x - 30\). Therefore, Ginny's initial error in identifying the sign of \(b\) led to an incorrect final result.

Examples

Understanding how to factor trinomials is crucial for solving quadratic equations in various real-life scenarios, such as modeling the trajectory of a launched projectile or optimizing dimensions to maximize the area of a garden. For instance, if you are designing a rectangular pool where the area depends on the width as \(-2x^2 + 10x + 12 = 0\), correctly factoring the expression allows you to easily find the roots, or the possible values for the width, which in this case would be \(x=6\) or \(x=-1\). Accuracy in every step ensures that the final design dimensions are correct and safe.

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