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Magazine Chapter 7: Problem 50
Solve each proportion. $$ \frac{s}{s-5}=\frac{s+5}{24} $$
Short Answer
Expert verified
The solutions are \(s = 25\) and \(s = -1\).
Step by step solution
01
Understand the Proportion
We start with the given proportion \( \frac{s}{s-5} = \frac{s+5}{24} \). Our goal is to solve for the variable \( s \).
02
Cross-Multiply
To eliminate the fractions, cross-multiply the terms. This means multiplying across the equal sign: \[ s \cdot 24 = (s - 5) \cdot (s + 5) \] This results in the equation, \[ 24s = (s - 5)(s + 5) \]
03
Apply the Difference of Squares
Recognize that the right side is a difference of squares, \((s - 5)(s + 5)\), which equals \(s^2 - 25\). Thus the equation becomes: \[ 24s = s^2 - 25 \]
04
Rearrange the Equation
Rearrange the terms to one side of the equation to set it to zero and form a quadratic equation: \[ s^2 - 24s - 25 = 0 \]
05
Solve the Quadratic Equation
Solve the quadratic equation \( s^2 - 24s - 25 = 0 \) using the quadratic formula \( s = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -24 \), and \( c = -25 \).Calculate the discriminant: \[ b^2 - 4ac = (-24)^2 - 4 \times 1 \times (-25) = 576 + 100 = 676 \]Since the discriminant is a perfect square, compute the solutions: \[ s = \frac{24 \pm \sqrt{676}}{2} = \frac{24 \pm 26}{2} \]The two potential solutions are:\( s_1 = \frac{50}{2} = 25 \) and \( s_2 = \frac{-2}{2} = -1 \).
06
Verify the Solutions
Check both solutions to ensure they don't cause any denominators to become zero, which would invalidate the solution. Plug back into the original proportion:- For \( s = 25 \): \( \frac{25}{25-5} = \frac{25+5}{24} \rightarrow \frac{25}{20} = \frac{30}{24} \rightarrow \frac{5}{4} = \frac{5}{4} \), which holds true.- For \( s = -1 \): \( \frac{-1}{-1-5} = \frac{-1+5}{24} \rightarrow \frac{-1}{-6} = \frac{4}{24} \rightarrow \frac{1}{6} = \frac{1}{6} \), which also holds true.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Difference of Squares
The difference of squares is a special algebraic identity that describes how two perfect squares can be subtracted. It is written in the form: \( (a - b)(a + b) = a^2 - b^2 \). This formula can be really handy in simplifying expressions and solving equations involving squares. In our exercise, we noticed that \((s-5)(s+5)\) fits this form perfectly, where \( a = s \) and \( b = 5 \).
- Recognizing the difference of squares allows us to replace the expression with \( s^2 - 25 \), which simplifies calculations considerably.
- This expression reveals a powerful shortcut for squaring differences and sums, often used in factoring and algebraic manipulation.
Cross-Multiplication
Cross-multiplication is an essential arithmetic technique often used to solve equations involving proportions. Proportions are statements that two ratios are equal, such as \( \frac{a}{b} = \frac{c}{d} \). To solve for one of the unknowns, you can eliminate the fractions by multiplying each side by the denominator of the opposite side.
- For the proportion in our exercise, \( \frac{s}{s-5} = \frac{s+5}{24} \), cross-multiplying involves multiplying \( s \) and 24, and \( (s-5) \) and \( (s+5) \). This results in the equation: \( 24s = (s-5)(s+5) \).
- It helps to "clear" the fractions and reduce the equation to a simpler form, making it easier to solve.
Quadratic Formula
When faced with a quadratic equation such as \( ax^2 + bx + c = 0 \), the quadratic formula is a reliable method to find its solutions. The quadratic formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula works for any quadratic equation, helping you uncover the two possible values for the variable. In our exercise, we used this formula to solve \( s^2 - 24s - 25 = 0 \).
- The coefficients were \( a = 1 \), \( b = -24 \), and \( c = -25 \). Calculating the discriminant, \( b^2 - 4ac \), helped determine the nature of the solutions (real and distinct, in this particular case, since we ended up with a positive discriminant).
- The solutions found were \( s = 25 \) and \( s = -1 \). These solutions were then verified to ensure they satisfied the original proportion.
