Problem 47 - Introductory Algebra for College Students

Question

Factor completely. $$10 x^{2}-40 x-600$$

Step-by-Step Solution

Answer

The factorized form of the given equation is \(10(x -10)(x + 6)\)

Step 1: Identify the Greatest Common Factor

Looking at the given quadratic equation \(10 x^{2}-40 x-600\), the common factors of 10, -40, and -600 is 10. Thus, we can factor out 10.

Step 2: Factor Out the Greatest Common Factor

Factor out 10 from each term in the given equation. Doing this will result in \(10(x^{2} - 4x -60)\).

Step 3: Factorize the Quadratic Equation

The quadratic equation \(x^{2} - 4x - 60\) can be factorized to \((x -10)(x + 6)\), using the fact that -4 is the sum of -10 and 6 and -60 is the product of -10 and 6.

Step 4: Write the Final Answer

The factorization of the original equation is then given by substituting \((x -10)(x +6) \) back into the equation after factoring out 10, which results in \(10(x -10)(x +6)\)

Key Concepts

Greatest Common Factor

When factoring a polynomial, one of the first steps is to identify the greatest common factor (GCF) of all the terms. The GCF is the largest number or expression that divides all terms in the polynomial without leaving a remainder. For example, in the expression given:

• 10 (\(x^2\)),
• -40 (x), and
• -600,

the greatest common factor is 10.
To find the GCF, list the factors of each coefficient and variable, then choose the largest factor common to all terms.
In this example, each number is divisible by 10, and there are no variables common to all terms, hence it's factored as:

• \(10x^2 - 40x - 600 = 10(x^2 - 4x - 60)\).

Factoring out the GCF simplifies the polynomial and makes further factorization steps easier.

Quadratic Equation

A quadratic equation is one that involves a polynomial of degree 2, typically expressed in the form

• $ax^2 + bx + c = 0$.

In the context of the problem, after factoring out the GCF, we are left with the quadratic equation

• $x^2 - 4x - 60 = 0$.

Solving quadratic equations can be accomplished using several methods, such as factoring, completing the square, or using the quadratic formula.
For the equation

• $x^2 - 4x - 60$,

the goal is to rewrite it as a product of binomials.
This transformation is possible if you can find two numbers that multiply to give the constant term

• (-60)

and add to give the coefficient of the linear term

• (-4).

Factoring Techniques

Factoring techniques are essential tools for simplifying polynomials and solving equations. In this exercise, we used a straightforward factoring technique known as the "zero product property" and trial and error to factor the quadratic.
Once we factor out the GCF and have the quadratic

• $x^2 - 4x - 60$,

we look for two numbers that, when multiplied, equal -60 and, when added, equal -4.
These numbers are -10 and 6. Using these numbers, the quadratic

• $x^2 - 4x - 60$

can be expressed as the product

• (x - 10)(x + 6).

After factoring, you multiply back to ensure the product matches the original quadratic. This confirmation step verifies your factorization is correct, maintaining the integrity of the solution.