Problem 194 - Elementary Algebra

Question

Factor. \(12 y^{2}-37 y+21\)

Step-by-Step Solution

Answer

(4y - 3)(3y - 7)

Step 1 - Identify coefficients

First, identify the coefficients of the quadratic equation. Here, we have the quadratic equation in the form of \(ax^2 + bx + c\) where: - \(a = 12\)- \(b = -37\)- \(c = 21\)

Step 2 - Find two numbers that multiply to \(a \cdot c\)

We need two numbers that multiply to \(12 \times 21 = 252\) and add up to -37. These two numbers are -9 and -28.

Step 3 - Rewrite the middle term

Rewrite the quadratic equation using these numbers to split the middle term: \(12y^2 - 9y - 28y + 21\)

Step 4 - Group terms

Group the terms to factor by grouping: \((12y^2 - 9y) + (-28y + 21)\)

Step 5 - Factor each group

Factor out the greatest common factor (GCF) from each group:\(3y(4y - 3) - 7(4y - 3)\)

Step 6 - Factor out the common binomial factor

Factor out the common binomial \((4y - 3)\): \[\begin{equation} (4y - 3)(3y - 7) \end{equation}\]

Step 7: Conclusion

The factored form of the quadratic equation is: \[\begin{equation}(4y - 3)(3y - 7)\end{equation}\]

Key Concepts

quadratic equations

A quadratic equation is a type of polynomial equation of degree 2. It is generally in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. In our example, \(12y^2 - 37y + 21\), we have:

• \(a = 12\)
• \(b = -37\)
• \(c = 21\)

The main goal in many problems is to factor the quadratic equation, which helps to solve for the variable \(y\) more easily. Factoring involves rewriting the quadratic in a form like this \((dy + e)(fy + g) = 0\). When you understand how to manipulate these equations, solving them becomes a lot easier.

factoring by grouping

Factoring by grouping is a method used to factor polynomials by separating them into groups. This technique is useful when a polynomial has four or more terms, but it can also apply to quadratic equations by first rewriting them. Here's the process we followed for our equation \(12y^2 - 37y + 21\):

• First, identify pairs of terms that can be factored together.
• Rewrite the middle term using numbers that multiply to the product of \(a\) and \(c\) but add up to \(b\).
• Group terms: \( (12y^2 - 9y) + (-28y + 21) \).

Now, we can factor each grouped term separately before combining the results. This step is crucial for simplifying our equation into a more manageable form.

common factors

Common factors are values or expressions that divide each reworked term of the polynomial exactly. In our factoring by grouping steps, we identified and factored them out:

• First group: \( 12y^2 - 9y = 3y(4y - 3) \)
• Second group: \( -28y + 21 = -7(4y - 3) \)

Notice that \(4y - 3 \) is a common factor. By factoring out \(4y - 3\), we combined the groups into: \( (4y - 3)(3y - 7) \). Finding and using common factors is a critical step for simplifying quadratic equations and makes it much easier to solve for \(y\). Factoring out these common terms reduces the polynomial and brings it to a product of two binomials.