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Magazine Chapter 5: Problem 10
Find the value of \(x\) in each proportion. a) \(\frac{x+1}{3}=\frac{10}{x+2}\) b) \(\frac{x-2}{5}=\frac{12}{x+2}\)
Short Answer
Expert verified
For part (a), possible solutions for \(x\) are -7 and 4; for part (b), solutions are 8 and -8.
Step by step solution
01
Identify the Equation for Part (a)
We are given the proportion \( \frac{x+1}{3}=\frac{10}{x+2} \). The task is to find the value of \(x\) that satisfies this equation.
02
Cross Multiply for Part (a)
To solve the proportion \( \frac{x+1}{3}=\frac{10}{x+2} \), we use cross multiplication. Multiply \((x+1)\) by \( (x+2) \) and \(3\) by \(10\): \((x+1)(x+2) = 3 \cdot 10\).
03
Expand and Simplify for Part (a)
Expand \((x+1)(x+2)\) to get \(x^2 + 3x + 2\). Thus, the equation becomes \(x^2 + 3x + 2 = 30\). Simplify it to form the quadratic equation: \(x^2 + 3x - 28 = 0\).
04
Solve the Quadratic Equation for Part (a)
Solve \(x^2 + 3x - 28 = 0\) using factoring: \((x+7)(x-4) = 0\). This gives two possible solutions for \(x\): \(x = -7\) or \(x = 4\).
05
Identify the Equation for Part (b)
We are given another proportion \( \frac{x-2}{5}=\frac{12}{x+2} \). We need to find the value of \(x\) for this equation.
06
Cross Multiply for Part (b)
Cross multiply in the proportion \( \frac{x-2}{5}=\frac{12}{x+2} \). Multiply \((x-2)\) by \((x+2)\) and \(5\) by \(12\): \((x-2)(x+2) = 5\cdot12\).
07
Expand and Simplify for Part (b)
Expand \((x-2)(x+2)\) to get \(x^2 - 4\). The equation becomes \(x^2 - 4 = 60\). Simplifying, we have \(x^2 - 64 = 0\).
08
Solve the Quadratic Equation for Part (b)
The equation \(x^2 - 64 = 0\) is a difference of squares. Factor it to \((x-8)(x+8)=0\), giving solutions \(x=8\) or \(x=-8\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cross Multiplication in Proportion Solving
Cross multiplication is a method used to solve equations that are set in a proportion format. A proportion is an equation that states that two ratios are equal, and it appears as a fraction equal to another fraction. To solve these equations, cross multiplication simplifies the process by eliminating the denominators.
Here's how cross multiplication works:
Here's how cross multiplication works:
- Given a proportion \( \frac{a}{b} = \frac{c}{d} \), cross multiply by multiplying \(a\) and \(d\) and \(b\) and \(c\).
- The equation \(a \times d = b \times c\) is formed.
Quadratic Equation Basics
Once you've used cross multiplication in a proportion, the next step often brings you to a quadratic equation. Quadratic equations are polynomial equations of second degree, usually in the form \(ax^2 + bx + c = 0\). Solving quadratic equations is essential in many mathematical problems.
Here are a few characteristics of quadratic equations:
Here are a few characteristics of quadratic equations:
- They have a squared term (\(x^2\)), a linear term (\(x\)), and a constant.
- They often appear after expanding products of binomials, such as \((x+1)(x+2)\) which expands to \(x^2 + 3x + 2\).
Factoring Methods to Solve Quadratic Equations
Factoring is one of the simplest methods to solve quadratics, if it's possible. This involves expressing the equation as a product of two binomials, which helps in finding the solutions for the variable \(x\).
The steps to factor effectively include:
The steps to factor effectively include:
- Start with identifying if the quadratic can be rewritten in the form \((x+m)(x+n) = 0\).
- Ensure that \(m\times n\) equals the constant term, and \(m+n\) equals the coefficient of \(x\).
