Problem 14 - Elementary Algebra

Question

Factor. $$ 25 x 2-10 x+1 $$

Step-by-Step Solution

Answer

The expression factors to \((5x - 1)^2\).

Step 1: Identify the Quadratic Expression

The expression given is a quadratic trinomial: \[ 25x^2 - 10x + 1 \] It is in the standard quadratic form \( ax^2 + bx + c \), where \( a = 25 \), \( b = -10 \), and \( c = 1 \).

Step 2: Check if it is a Perfect Square Trinomial

A perfect square trinomial is of the form \((ax)^2 - 2abx + b^2\). Let's check if the given expression fits this pattern:1. The first term is \((5x)^2 = 25x^2\).2. The last term is \(1^2 = 1\).3. The middle term should be \(-2(5x)(1) = -10x\), which matches the given middle term. Thus, \(25x^2 - 10x + 1\) is a perfect square trinomial.

Step 3: Write the Factored Form

Since \(25x^2 - 10x + 1\) is a perfect square trinomial, we can write it as:\[(5x - 1)^2\].

Step 4: Verify the Solution

Expand \((5x - 1)^2\) to check:\[(5x - 1)(5x - 1) = 25x^2 - 5x - 5x + 1 = 25x^2 - 10x + 1\].The original expression is obtained, confirming the factorization is correct.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of quadratic trinomial that can be expressed as the square of a binomial. To identify a perfect square trinomial, we look for the following properties:

• The first term is a perfect square.
• The last term is a perfect square.
• The middle term is twice the product of the square roots of the first and the last terms, with an appropriate sign.

In our exercise, the quadratic expression is \(25x^2 - 10x + 1\). Let's break it down:

• The first term, \(25x^2\), is \((5x)^2\).
• The last term, \(1\), is \(1^2\).
• The middle term, \(-10x\), matches with \(-2(5x)(1)\).

Since all conditions are satisfied, \(25x^2 - 10x + 1\) is indeed a perfect square trinomial, and can be written as \((5x - 1)^2\).

Quadratic Trinomial

A quadratic trinomial is a polynomial with three terms where the highest degree is two, generally expressed in the form \(ax^2 + bx + c\). Each coefficient \(a\), \(b\), and \(c\) plays a role:

• \(a\) is the coefficient of the square term \(x^2\).
• \(b\) is the coefficient of the linear term \(x\).
• \(c\) is the constant term.

In our case, the trinomial \(25x^2 - 10x + 1\) has:

• \(a = 25\)
• \(b = -10\)
• \(c = 1\)

Recognizing a quadratic trinomial's structure aids in approaches to factorization, as certain patterns, like perfect square trinomials, can simplify the process.

Factorization Verification

After factoring a quadratic expression, it's crucial to verify the factorization by expansion to ensure its correctness. Verification involves expanding the factored expression and checking whether it returns to the original quadratic trinomial.In the exercise, once we identified \(25x^2 - 10x + 1\) as a perfect square trinomial, we factored it into \((5x - 1)^2\).Let's verify this factorization:

• Expand \((5x - 1)^2\):

\[(5x - 1)(5x - 1) = 25x^2 - 5x - 5x + 1 = 25x^2 - 10x + 1\]As seen, expanding \((5x - 1)^2\) returns us to the original expression. This confirms our factorization is correct and accurate.

Standard Quadratic Form

The standard quadratic form is the common structure of quadratic equations shown as \(ax^2 + bx + c\). This form makes it easier to identify patterns and determine methods of solving or factoring the quadratic expression.Understanding the components:

• \(a\) is the coefficient that affects the width and direction of the parabola generated by the quadratic.
• \(b\) influences the placement of the vertex horizontally.
• \(c\) adjusts the vertical position of the graph.

For the equation \(25x^2 - 10x + 1\), recognizing its standard form helps identify it as a quadratic trinomial. This understanding guides the approach to factoring or solving the equation.Standard form is especially valuable in identifying specific types of trinomials, like perfect square trinomials, that simplify the factorization process. It provides a methodological way to approach and simplify quadratic equations.