Problem 53 - Algebra and Trigonometry

Question

Factor the expression completely. $$ t^{2}-6 t+9 $$

Step-by-Step Solution

Answer

The expression factors to \((t-3)^2\).

Step 1: Identify the Quadratic Expression

The expression given is a quadratic equation of the form \( t^2 - 6t + 9 \). Our goal is to factor this expression by finding two binomials that multiply together to give this expression.

Step 2: Understand the Perfect Square Form

Notice that the expression \( t^2 - 6t + 9 \) resembles the formula for a perfect square trinomial, which is \( (a-b)^2 = a^2 - 2ab + b^2 \).

Step 3: Solve Using Perfect Square Trinomial

Let's compare \( t^2 - 6t + 9 \) with the perfect square trinomial formula. Here, \( a^2 = t^2 \), which means \( a = t \), and \( b^2 = 9 \), so \( b = 3 \). Also, check that \(-2ab = -6t \), confirming \(-2 \times t \times 3 = -6t \).

Step 4: Factor the Expression

Since \( t^2 - 6t + 9 \) is a perfect square trinomial, we can factor it as \( (t-3)^2 \).

Step 5: Verify the Factorization

To ensure the factorization is correct, expand \( (t-3)^2 \) to see if it results in the original expression: \( (t-3)(t-3) = t^2 - 3t - 3t + 9 = t^2 - 6t + 9 \). This matches the original expression, confirming that the factorization is accurate.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression. You can recognize it when you see a pattern similar to \((a - b)^2 = a^2 - 2ab + b^2\). This form means the expression can be easily rewritten as the square of a binomial.
In our example, we have the expression \(t^2 - 6t + 9\). Let's break this down:

• Compare it to \(a^2 - 2ab + b^2\): Here, \(a^2 = t^2\), so \(a = t\).
• The middle term, \(-6t\), gives \(-2ab = -6t\), leading to \(b = 3\).
• The last term, \(9\), equals \(b^2\), confirming \(b = 3\).

So, we have verified that \(t^2 - 6t + 9\) fits perfectly into this pattern,
making it a perfect square trinomial.
This understanding allows you to directly factor the expression into \((t-3)^2\).
This approach simplifies the factorization process, making it much more straightforward.

Quadratic Equation

A quadratic equation is any equation involving the square subseript or second degree of a variable. It typically takes the form \(ax^2 + bx + c = 0\). In our exercise,
the quadratic expression is \(t^2 - 6t + 9\). In general terms, you deal with three components:

• \(a\), the coefficient in front of \(t^2\): Controls how wide or narrow the parabola appears.
• \(b\), the linear coefficient: Affects the axis of symmetry for the parabola.
• \(c\), the constant term: Represents the graph's vertical shift.

Quadratic equations are everywhere in math and science. They're essential tools
for modeling parabolic motions like the trajectory of a ball. Understanding their structure
helps solve real-world problems effectively. The ability to factor quadratic equations
also makes finding their roots—a core aspect—efficient.

Binomial Factorization

Binomial factorization focuses on expressing a quadratic expression as a product
of two binomials. For example, our original exercise involves the expression \(t^2 - 6t + 9\). It's factored simply as \((t-3)(t-3)\) or \((t-3)^2\).

This straightforward factorization comes from recognizing the expression is a perfect square trinomial. The steps for binomial factorization include:

• Identify if the quadratic expression matches a known pattern like the perfect square.
• Rewrite the expression by finding values of \(a\) and \(b\) in \((a-b)^2\) form.
• Verify correctness by expanding the binomials back to the original expression.

This method not only simplifies solving but also helps in quickly understanding
the characteristics of quadratic expressions. Spotting such patterns transforms
potentially complex solutions into easy and manageable processes. This skill is essential
in algebra and vital for succeeding in mathematics.