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Magazine Chapter 11: Problem 69
Solve. $$ \frac{x}{x-1}-6 \sqrt{\frac{x}{x-1}}-40=0 $$
Short Answer
Expert verified
The solution is x = \( \frac{100}{99} \).
Step by step solution
01
- Substitute Variable
Let us introduce a substitution to simplify the equation. Let \( y = \frac{x}{x-1} \). This changes our equation to: \( y - 6\text{鈭殅y - 40 = 0 \)
02
- Solve the Quadratic Equation
We have a quadratic equation in terms of \( y \). Let's solve it: \( y - 6\text{鈭殅y - 40 = 0 \). Let \( \text{鈭殅y = t \), which rewrites our equation as: \( t^2 - 6t - 40 = 0 \)
03
- Use the Quadratic Formula
Solve the equation \( t^2 - 6t - 40 = 0 \) using the quadratic formula: \( t = \frac{-b \text{卤} \text{鈭殅(b^2 - 4ac)}{2a} \), where \( a = 1 \), \( b = -6 \), and \( c = -40 \). Calculate the roots: \( t = \frac{6 \text{卤} \text{鈭殅(36 + 160)}{2} \) which simplifies to: \( t = \frac{6 \text{卤} \text{鈭殅196}{2} \), and finally simplifies to: t = 10 or t = -4
04
- Back Substitute
Recall \( t = \text{鈭殅y \). Then we have \( \text{鈭殅y = 10 \) or \( \text{鈭殅y = -4 \). Since \( \text{鈭殅y \) cannot be negative, discard \( -4 \). Thus, \( \text{鈭殅y = 10 \) implies \( y = 100 \).
05
- Substitute Back
Substitute \( y = 100 \) back into \( y = \frac{x}{x-1} \): \( \frac{x}{x-1} = 100 \). This gives us the equation: \( x = 100(x-1) \). Simplify it: \( x = 100x - 100 \). Then solve for \( x \) : \( 100 = 99x \), which gives \( x = \frac{100}{99} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equations
Quadratic equations are a central concept in algebra. They鈥檙e equations of the form \( ax^2 + bx + c = 0 \). Here, \( a, b, \) and \( c \) are constants. These equations have at most two solutions. The solutions can be real or complex numbers. To solve quadratic equations, several methods are available:
- Factoring: This works for simple quadratic equations, where you can break it down into two binomial expressions.
- Quadratic Formula: This is a universal method for solving all quadratic equations. It鈥檚 given by the formula: \( x = \frac{-b \text{卤} \text{鈭�(b^2 - 4ac)}}{2a} \).
- Completing the Square: This involves manipulating the equation to form a perfect square trinomial.
Substitution Method
The substitution method is a powerful tool in algebra for simplifying complex equations. It involves replacing a part of the equation with a new variable. This can make solving the equation easier. Here鈥檚 a breakdown:
- Identify a Complex Part: Find a part of the equation that is difficult to work with.
- Introduce a New Variable: Replace the identified part with a new variable, say \( y \).
- Solve for the New Variable: Simplify and solve the equation in terms of the new variable.
- Back-Substitute: Reinsert the original expression to find the solution to the original equation.
Solving Equations
Solving equations is a fundamental skill in algebra. There are different strategies depending on the type of equation:
Remember to always verify your solutions by substituting them back into the original equation to ensure they satisfy it fully.
- Linear Equations: These can be solved by isolating the variable on one side of the equation.
- Quadratic Equations: Deploy methods like factoring, using the quadratic formula, or completing the square as discussed earlier.
- Substitution: For equations that can be simplified by introducing a new variable.
- Graphing: Sometimes, visualizing the solutions by plotting graphs can be very helpful.
Remember to always verify your solutions by substituting them back into the original equation to ensure they satisfy it fully.
