Problem 51 - Calculus: One and Several Variables

Question

Write the expression in factored form. \(x^{2}-10 x+25\).

Step-by-Step Solution

Answer

The factored form of the expression \(x^{2}-10x+25\) is \((x - 5)^2\).

Step 1: Identify the General Form

The given quadratic expression is in the general form of \(ax^2 + bx + c\), where \(a = 1\), \(b = -10\), and \(c = 25\). This corresponds to the quadratic expression \(x^{2}-10x+25\).

Step 2: Recognize the Pattern

A perfect square trinomial can be recognized if the square root of the first term and last term add up or subtract to have the middle term coefficient. The square root of \(x^{2}\) is \(x\) and the square root of \(25\) is \(5\). The double of the product of these two square roots gives us \(10\). This equals the absolute value of \( -10\), the coefficient of the middle term. Thus, we have a perfect square trinomial.

Step 3: Write in Factored Form

The general formula for a perfect square trinomial is \((a-b)^2 = a^2 - 2ab + b^2\). Since \(a = x\) and \(b = 5\), we can rewrite the equation as \((x - 5)^2\).

Key Concepts

Perfect Square Trinomial

Understanding what a perfect square trinomial is can make factoring much easier! These trinomials have a specific structure that allows them to be expressed as the square of a binomial. Let's break it down.

A trinomial is a quadratic expression that has three terms. But not all trinomials are perfect squares. A perfect square trinomial is of the form:

• \(a^2 - 2ab + b^2\)
• \(a^2 + 2ab + b^2\)

To identify it, check these things:

• Both the first and last terms should be perfect squares.
• The middle term should be twice the product of the square roots of the first and last terms.

In the expression \(x^{2}-10x+25\):

• The square root of \(x^2\) is \(x\).
• The square root of \(25\) is \(5\).
• The middle term \(-10x\) is \(-2 \times x \times 5\), satisfying the conditions of a perfect square trinomial.

Knowing this pattern helps you recognize and factor these trinomials easily!

Factored Form

Factoring a quadratic expression means rewriting it as a product of simpler expressions. This is especially useful in algebra, as it simplifies equations and helps to find their roots.

For a trinomial like \(x^{2}-10x+25\), once you recognize it as a perfect square trinomial, factoring becomes straightforward. The general formula used is:

• \((a - b)^2 = a^2 - 2ab + b^2\)
• \((a + b)^2 = a^2 + 2ab + b^2\)

In this case, since \(a = x\) and \(b = 5\), the factored form is:

• \((x - 5)^2\)

This means the expression \(x^{2}-10x+25\) can be rewritten as \((x-5)(x-5)\). This form is more digestible and can easily tell us the roots of the equation when set to zero. Factored form reveals much about a quadratic's behavior and is foundational in solving equations.

Quadratic Expression

A quadratic expression is a polynomial where the highest exponent of the variable is 2. Typically, it follows the structure \(ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. Quadratic expressions appear in various mathematical contexts and have unique properties worth noting.

Characteristics include:

• Parabolic graph shape when plotted.
• Symmetrical property around its vertex line.

For \(x^{2}-10x+25\):

• The coefficient \(a = 1\) shapes the curve's width.
• The term \(-10x\) moves the parabola horizontally.
• The constant \(25\) shifts it vertically.

Quadratic expressions are the heart of many algebra problems and solutions often involve factoring, graphing, or using the quadratic formula.