Problem 219 - Elementary Algebra

Question

Factor. \(100 x^{2}-20 x+1\)

Step-by-Step Solution

Answer

The quadratic \(100x^2 - 20x + 1\) factors to \((10x - 1)^2\).

Step 1: Identify coefficients

The quadratic expression is of the form \(ax^2 + bx + c\).Here, \(a = 100\), \(b = -20\), and \(c = 1\).

Step 2: Determine if the quadratic is a perfect square trinomial

A quadratic expression can be factored into a perfect square trinomial if it is of the form \((mx + n)^2\).For this expression to be a perfect square trinomial: \((mx + n)^2 = m^2x^2 + 2mnx + n^2\).Set the given coefficients equal to the expanded perfect square trinomial: \(100x^2 - 20x + 1\) = \((10x - 1)^2\).

Step 3: Verify by expanding

Expand the factor \((10x - 1)(10x - 1) = 10x \times 10x + 10x \times (-1) + (-1) \times 10x + (-1)(-1)\) = \(100x^2 - 10x - 10x + 1 = 100x^2 - 20x + 1\).Since the expanded form matches the original quadratic expression, \((10x - 1)^2\) is a valid factorization.

Step 4: Write the final factored form

The final factored form of the quadratic \(100x^2 - 20x + 1\) is \((10x - 1)^2\).

Key Concepts

Perfect Square Trinomials

Perfect square trinomials are special quadratic expressions that can be factored into a binomial squared. They follow the form \((ax + b)^2 = a^2x^2 + 2abx + b^2\). This means the trinomial has two identical binomial factors.
By recognizing perfect square trinomials, you can simplify the factoring process. For example, the expression given in the problem is \[100x^2 - 20x + 1\].
Notice it fits the pattern:

• The coefficient of the quadratic term is 100, and \(10^2 = 100\).
• The middle term \(-20x\) is \(-2 * 10 * 1\).
• The constant term is \(1\).

This shows that \[100x^2 - 20x + 1\] is a perfect square trinomial and can be factored as \((10x - 1)^2\).

Quadratic Expressions

Quadratic expressions are algebraic expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The expression has three main terms:

• The \(ax^2\) term is the quadratic term.
• The \(bx\) term is the linear term.
• The constant term \(c\) is the independent term.

Factoring quadratic expressions involves finding two binomials that, when multiplied together, yield the original quadratic. Recognizing patterns like perfect square trinomials can make this process faster.
In our exercise, the given quadratic expression is \[100x^2 - 20x + 1\]. Identifying coefficients \(a = 100\), \(b = -20\), and \(c = 1\) helps you to see its structure and decide on the best factoring method.

Factored Form

Factoring a quadratic expression implies rewriting it as the product of two binomials. Generally, the factored form of a quadratic \(ax^2 + bx + c\) is \((mx + n)(px + q)\).
In cases of perfect square trinomials, the factored form is simpler. It becomes a binomial squared, \((px + q)^2\).
The steps to convert the quadratic given in our problem to its factored form are straightforward:

• First, identify the coefficients: \(a = 100\), \(b = -20\), and \(c = 1\).
• Recognize it fits the perfect square trinomial pattern \[ (ax + b)^2 = a^2x^2 + 2abx + b^2\].
• Verify by expanding \((10x - 1)(10x - 1) = 100x^2 - 20x + 1\).

After verification, you write the factored form as \((10x - 1)^2\).
So, the factored form of \[100x^2 - 20x + 1\] is \((10x - 1)^2\), showcasing one of the neatest ways to simplify quadratic expressions.