Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-14: Solve the equation by factoring. $$ 4 x^{2}-72 x+324=0 $$

Step-by-Step Solution

Answer

The solution to the equation is \( x = 9 \).

Step 1: Identify a Common Factor

First, look for a common factor in all the terms of the equation: \[ 4x^2 - 72x + 324 = 0 \]Each term is divisible by \(4\). Factor \(4\) out of the equation:\[ 4(x^2 - 18x + 81) = 0 \]

Step 2: Apply the Zero Product Property

Divide both sides of the equation by \(4\) to simplify:\[ x^2 - 18x + 81 = 0 \].Now you can focus on solving the quadratic inside the parentheses.

Step 3: Look for a Perfect Square

Recognize that the quadratic \( x^2 - 18x + 81 \) follows the pattern of a perfect square trinomial:\[ (x - 9)^2 = 0 \].The expression is a perfect square because \(\left(-18/2\right)^2 = 81\).

Step 4: Solve for the Variable

Apply the square root property to solve:\[ (x - 9)^2 = 0 \].Take the square root of both sides:\[ x - 9 = 0 \], thus \( x = 9 \).

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation that have the form \( ax^2 + bx + c = 0 \). This means they involve a variable, often \( x \), raised to the power of two, making them second-degree equations. Quadratics are fundamental in algebra because they describe parabolas when graphed, and they appear in various real-life scenarios like projectile motion and optimization problems.

To solve a quadratic equation, you can use different methods such as:

• Factoring: Finding two binomials that multiply together to give the original quadratic.
• Quadratic formula: Utilizing the formula \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \) to solve for \( x \).
• Completing the square: Turning the quadratic into a perfect square trinomial for easier solving.

In the exercise, we use factoring by recognizing the quadratic as a perfect square trinomial. This simplifies the process significantly, as it reduces the equation to a form we can easily solve.

Zero Product Property

The Zero Product Property is a useful tool when solving equations that involve factoring. It states that if the product of two numbers is zero, at least one of the numbers must be zero. This concept is crucial because it means if \( a \times b = 0 \), then either \( a = 0 \) or \( b = 0 \).

Applying this concept involves:

• Factoring the equation such that it can be expressed as a product of factors set equal to zero.
• Setting each factor to zero individually and solving for the variable.

In the given exercise, after factoring out \( 4 \), we are left with \( (x - 9)^2 \). Applying the Zero Product Property here is straightforward, leading us directly to the solution because squaring zero gives us zero again. Therefore, the solution reduces to solving \( x - 9 = 0 \), quickly isolating \( x \).

Perfect Square Trinomial

A perfect square trinomial is a special form of quadratic equation where the quadratic expression can be written as the square of a binomial. This means that the trinomial \( ax^2 + bx + c \) can be expressed in the form \( (mx + n)^2 \). Understanding this property makes solving certain quadratics simpler, as they can be easily factored.

To recognize a perfect square trinomial:

• Identify that the first term \( ax^2 \) and the last term \( c \) are perfect squares.
• Ensure that the middle term \( bx \) is twice the product of the square roots of the first and last terms.

In our exercise, the expression \( x^2 - 18x + 81 \) is a perfect square trinomial. This is verified because \((-18/2)^2 = 81\), matching the format \((x - 9)^2\). This recognition allows for a quick solution using simple algebraic principles, illustrating the power of understanding perfect squares in algebra.