Question

Which of the following is a solution to the equation below? \((x-3)^{2}-81=0\) (A) 12 (B) 11 (C) 9 (D) 8

Short answer

The solution to the equation is 12 (option A).

Step-by-step solution

Expand the Equation

Start by expanding the equation \((x-3)^{2}-81=0\). To expand \((x-3)^{2}\), apply the binomial theorem: \((x-3)^{2} = x^2 - 6x + 9\). Thus, the equation becomes \(x^2 - 6x + 9 - 81 = 0\).

Simplify the Equation

Combine like terms in the equation \(x^2 - 6x + 9 - 81 = 0\). Simplifying the constants, the equation becomes \(x^2 - 6x - 72 = 0\).

Factor the Quadratic Equation

Look for two numbers that multiply to -72 and add up to -6. Those numbers are -12 and 6. Therefore, factor the equation as \((x - 12)(x + 6) = 0\).

Solve for x

Set each factor equal to zero to solve for \(x\). Thus, \(x - 12 = 0\) gives \(x = 12\) and \(x + 6 = 0\) gives \(x = -6\).

Verify the Correct Answer

The problem asks for which given option is a solution. The available options were: (A) 12, (B) 11, (C) 9, (D) 8. The correct value that solves the equation from the options is \(x = 12\).

Key concepts

Solving Quadratic Equations

Quadratic equations are polynomials that include a variable squared ( x^2 ). These types of equations usually take the form x^2 + bx + c = 0 . Solving a quadratic equation means finding the values of x that make the equation true. There are various methods to solve them, including factoring, using the quadratic formula, and completing the square. In our exercise, the expanded form is x^2 - 6x - 72 = 0 .
To solve, we need to find values for x that satisfy the equation. Placing the equation into standard form helps identify potential methods for solving it. Once in standard form, look at the possibilities for factoring to make finding the solution steps more manageable.

Factoring Equations

Factoring is a crucial technique for solving equations that involve polynomials. The main goal when factoring is to break down a complex expression into simpler factors that can be set to zero. For example, when you have x^2 - 6x - 72 = 0 , you want to find two numbers that multiply to -72 while adding up to -6 .
These two numbers are -12 and 6 . Thus, you can write the equation in a factored form as (x - 12)(x + 6) = 0 . This method allows us to solve the quadratic equation by setting each factor to zero, providing potential solutions for x . It’s a straightforward approach when the equation is easily factorable, as it was in this instance.

Binomial Theorem Application

The binomial theorem is a formula used to expand expressions that are raised to a power, such as (x-3)^2 . It’s instrumental because it simplifies the process of expanding expressions without multiplying them out directly. The binomial theorem states that (a+b)^n can be expanded based on combinations and powers of a and b . For (x-3)^2 , use this approach to transform it into x^2 - 6x + 9 .
Applying the binomial theorem makes complex expansions more manageable and reveals the quadratic equation's structure, x^2 - 6x + 9 - 81 = 0 , leading to simpler, more efficient problem-solving. Understanding how to apply this theorem enhances both algebraic understanding and the ability to manipulate expressions.

PSAT Preparation

Preparing for the PSAT involves building a strong foundation in math concepts, especially algebra and geometry. Quadratic equations often appear on the PSAT, along with other types of algebraic expressions. Practicing problems that require expanding, factoring, and solving quadratic equations can significantly improve your score.
• Familiarize yourself with key algebra techniques, like factoring and using the quadratic formula.
• Work on practice problems to become comfortable with various equation forms.
• Time management skills are crucial, so practice under timed conditions to enhance your efficiency.
Additionally, leveraging resources such as online tutorials, practice tests, and PSAT prep books can further reinforce your understanding and boost your confidence for the exam.