Problem 51 - Elementary Algebra

Question

A student is using the \(a c\) method to factor \(6 x^{2}-11 x-35\). Since \(a=6, b=-11, c=-35\), and \(a c=-210\), the student needs to find two numbers whose sum is \(-11\) and whose product is \(-210\). The student does not identify the factors paired with \(1,2,3\), and 5 because she thinks that the sum of these factors cannot be \(-11\). Is the student correct? Explain.

Step-by-Step Solution

Answer

The student is incorrect; the correct factors are 10 and -21.

Step 1 - Define the problem

Identify the given quadratic equation and the values of its coefficients: \[6x^{2} - 11x - 35\]Here, \[a = 6, b = -11, c = -35\]Calculate the product of \[a\] and \[c\]: \[ac = 6 \times -35 = -210\].

Step 2 - State the goal

The goal is to find two numbers that multiply to \(-210\) and add to \(-11\).

Step 3 - List factor pairs

List all pairs of factors of \(-210\): \[ (1, -210), (-1, 210), (2, -105), (-2, 105), (3, -70), (-3, 70), (5, -42), (-5, 42), (6, -35), (-6, 35), (7, -30), (-7, 30), (10, -21), (-10, 21), (14, -15), (-14, 15).\]

Step 4 - Check sums of factor pairs

Check each pair to find the one that sums to \(-11\): \[(1 - 210) = -209\], \[(-1 + 210) = 209\], \[(2 - 105) = -103\], \[(-2 + 105) = 103\], \[(3 - 70) = -67\], \[(-3 + 70) = 67\], \[(5 - 42) = -37\], \[(-5 + 42) = 37\], \[(6 - 35) = -29\], \[(-6 + 35) = 29\], \[(7 - 30) = -23\], \[(-7 + 30) = 23\], \[(10 - 21) = -11\].

Step 5 - Validate student’s claim

Determine if the student’s claim about factor pairs involving \[1, 2, 3\], and \[5\] is correct. The student missed the pairs that correctly sum to \[-11\], specifically \[10\text {-} 21\]. Hence, the student's assumption is incorrect.

Step 6 - Formulate final answer

The correct factor pairs are \[(10, -21)\] as \[10 + (-21) = -11\] and \[10 \times (-21) = -210\].

Key Concepts

ac method

The ac method is a powerful technique used to factor quadratic equations of the form \(ax^2 + bx + c\). This method is particularly useful when the quadratic equation does not factor easily using simple methods. Using the numbers \(a\), \(b\), and \(c\) from the equation, you first multiply \(a\) and \(c\). In our case, for the equation \(6x^2 - 11x - 35\), we have \(a = 6\) and \(c = -35\). Multiplying these gives us \(ac = 6 * (-35) = -210\).
The next step is to find two numbers that multiply to this product and add up to \(b\), which is \(-11\). This concept is central to the ac method, as it helps us rewrite the middle term \(bx\) into two terms that can be grouped and factored further. If executed properly, the ac method transforms the complex quadratic equation into a product of binomials, revealing its factors clearly and concisely.

factor pairs

Factor pairs are pairs of numbers that, when multiplied together, yield a specific product. In the context of the ac method, we are interested in factor pairs that relate to the product of \(a\) and \(c\). For example, in the quadratic equation \(6x^2 - 11x - 35\), the product of \(a\) and \(c\) is \(-210\). We need to list out all possible pairs whose multiplication equals \(-210\).
This can include pairs such as \((1, -210), (-1, 210), (2, -105), (-2, 105)\), and so forth. One of these pairs must also add up to \(-11\), the coefficient of \(x\).
Checking these pairs is crucial; each pair needs to meet both the multiplication and addition requirement. As seen in the solution, the correct pair is \((10, -21)\), since \(10 + (-21) = -11\). Understanding how to correctly identify and verify these pairs is key to correctly factoring quadratic equations using the ac method.

sum and product of factors

In factoring quadratic equations, finding two numbers that fit both a sum and product criteria is often the trickiest part. Reiterating our specific example of \(6x^2 - 11x - 35\), recall we need two numbers that multiply to \(-210\) and sum to \(-11\).
We systematically check each factor pair of \(-210\) to see if it satisfies these conditions. This involves simple arithmetic checks: summing and multiplying the factor pairs until we stumble upon \((10, -21)\).
The reason these criteria are crucial is because they allow us to split the middle term of the quadratic equation, \(-11x\), into two terms that can be grouped and factored further. This splitting, made possible by meeting both the sum and product conditions, transforms the quadratic equation into a form that's easy to factor, leading to the final solution in a logical sequence. By mastering this technique, you're better equipped to tackle a wide range of quadratic equations with confidence.