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Chapter 10: Problem 14

Use the zero-product property to solve the equation. $$ (x-7)^{2}=0 $$

Short Answer

Expert verified
The solution to the equation is x = 7.

Step by step solution

01

Identify the factors

The factors of the equation are (x-7)^2. Here, the same factor appears twice, and it equates to zero.
02

Apply the Zero-Product Property

According to the zero-product property, if (x-7)^2 = 0, then (x-7) must be zero. So, set each factor equal to zero and solve for x.
03

Solve for x

Setting the factor (x-7) equal to zero gives the equation x-7 = 0, which simplifies to x = 7.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Equations
Algebraic equations are mathematical statements that use variables and constants to express relationships. They typically involve operations such as addition, subtraction, multiplication, and division. In an algebraic equation, the goal is often to solve for the unknown variable.
For example, in the equation \( (x-7)^2 = 0 \), \( x \) is the variable that we are trying to find. The equation structure shows a squared binomial, meaning we have the same expression repeated under multiplication.
  • Variables: Symbols representing unknown values, often denoted by letters like \( x \).
  • Constants: Known values, such as numbers like 7 in \( (x-7)^2 \).
  • Expressions: Combinations of variables and constants that form parts of an equation.
  • Equations: Mathematical sentences stating two expressions are equal, as seen in our example.
Understanding and identifying these components helps us effectively approach solving equations.
Solving Quadratic Equations
Solving quadratic equations is a fundamental skill in algebra. Quadratic equations are typically expressed in the form \( ax^2 + bx + c = 0 \). However, they can also appear in other forms like \( (x-7)^2 = 0 \), which is a simplified version due to factoring.
The equation \( (x-7)^2 = 0 \) includes a repeated factor, which simplifies the process of finding the solution. Here, the zero-product property is applied directly.
Steps to solve:
  • Identify the form: Recognize it as a quadratic due to the squaring of a term.
  • Apply the Zero-Product Property: If \( a \cdot b = 0 \), then \( a = 0 \) or \( b = 0 \).
  • Solve for \( x \): When \( (x-7)^2 = 0 \), this means \( x-7 = 0 \).
  • Simplify: Solve the linear equation to find \( x = 7 \).
These steps ensure clarity by breaking the process into manageable parts, which precisely guide one to the solution.
Mathematical Properties
Mathematical properties are pivotal tools in solving equations and understanding algebraic expressions. One of these crucial properties is the Zero-Product Property. This property states that for any real numbers \( a \) and \( b \), if \( a \cdot b = 0 \), then either \( a = 0 \), \( b = 0 \), or both.
In the example \( (x-7)^2 = 0 \), the Zero-Product Property reveals its value by simplifying the equation solving process.
Key aspects:
  • Utility: It allows for breaking down complex quadratic expressions.
  • Application: Focuses on finding the values that make each factor zero.
  • Simplicity: Reduces the problem to solving linear equations.
By utilizing the Zero-Product Property, mathematicians can systematically solve quadratic equations efficiently, proven by its application in both basic algebra and more complex mathematical scenarios.

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Most popular questions from this chapter

Solve the equation. \(|2 x-5|+7=16\)

Simplify the expression. $$\frac{-6 \sqrt{12}}{\sqrt{4}}$$

Which of the following is a correct factorization of \(72 x^{2}-24 x+2 ?\) (A) \(-9(3 x-1)^{2}\) (B) \(8\left(9 x-\frac{1}{2}\right)^{2}\) (C) \(8\left(3 x-\frac{1}{2}\right)\left(3 x-\frac{1}{2}\right)\) (D) \(-8\left(3 x-\frac{1}{2}\right)^{2}\)

Decide whether or not the ordered pair is a solution of the system of linear equations. $$\begin{aligned} &2 x+6 y=22\\\ &-x-4 y=-13 \quad(-5,-2) \end{aligned}$$

Which one of the following equations cannot be solved by factoring with integer coefficients? (A) \(12 x^{2}-15 x-63=0\) (B) \(12 x^{2}+46 x-8=0\) (C) \(6 x^{2}-38 x-28=0\) (D) \(8 x^{2}-49 x-68=0\)
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