Problem 69 - Introductory Algebra for College Students

Question

Use the negative of the greatest common factor to factor completely. $$-5 x^{2}+50 x-45$$

Step-by-Step Solution

Answer

-5(x-9)(x-1)

Step 1: Identify the common factors

In \(-5x^{2} + 50x - 45\), the common factor is 5. This can be found by looking at each term individually and identifying a number that can be multiplied to give the coefficient of each term.

Step 2: Factor out the common factor

Since the problem is asking for the negative of the greatest common factor, we actually factor out \(-5\). When you factor out a -5 from each term, the equation becomes: \(-5(x^{2} - 10x + 9)\).

Step 3: Factor the quadratic expression

Now we need to factor the quadratic expression \(x^{2} - 10x + 9\). This can be done by finding two numbers that multiply to 9 and add up to -10. These numbers are -1 and -9. Therefore, the quadratic expression factorizes to \((x-9)(x-1)\).

Step 4: Combine steps 2 and 3

Combining the results from step 2 and 3 gives us the completely factored form of the given expression. Hence, \(-5 x^{2}+50 x-45 = -5(x-9)(x-1)\).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a key concept in algebra that simplifies expressions. The GCF is the largest factor shared by all terms of a given expression. In simpler terms, it is the biggest number or algebraic factor that divides all the terms without leaving a remainder. Finding the GCF helps in factoring expressions, making them easier to solve or simplify.

To find the GCF of the terms in an expression like

• -5x² + 50x - 45,

we look at each coefficient (the number in front of each variable) first. Here, the coefficients are -5, 50, and -45. The greatest number that can evenly divide each of these coefficients is 5. However, since the problem specifies the negative GCF, we're interested in

• -5 as the GCF.

Factoring out the negative GCF simplifies the expression, allowing us to see its basic components more clearly.

Quadratic Equations

Quadratic equations are algebraic expressions of the form

• ax² + bx + c,

where a, b, and c are constants, and x represents an unknown variable. They appear frequently in algebra and have a characteristic U-shaped graph called a parabola.

To solve quadratic equations like

• x² - 10x + 9,

we factor them into simpler expressions. This is done by finding two numbers that both add up to the middle term, -10 in this case, and multiply to the constant term, which is 9. Here, those numbers are -1 and -9. Factoring the expression gives us two linear factors: (x - 9) and (x - 1).

Once factored, you can either solve for x or use these factors to multiply back and verify completeness of factorization.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. In algebra, expressions like

• -5x² + 50x - 45

are factored to identify simpler components. Factoring is breaking down an expression into products of other expressions that when multiplied together, give the original expression.

When working with algebraic expressions, managing terms and identifying patterns are crucial skills. Factoring involves looking for ways to express each part of the expression in terms of a common factor or other algebraic relationships. In more complex expressions, recognizing a pattern like a quadratic can help guide you through factoring steps. Once factored, expressions are easier to manipulate or use in further calculations, such as solving equations or modeling real-world scenarios.

Understanding how algebraic expressions can be rewritten in a more straightforward manner is essential for solving many algebra problems, including quadratic equations and finding roots or solutions.