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Magazine Chapter 1: Problem 91
Find all real solutions of the equation. $$\frac{10}{x}-\frac{12}{x-3}+4=0$$
Short Answer
Expert verified
The real solutions are \(x = 5\) and \(x = -1.5\).
Step by step solution
01
Identify the Common Denominator
The equation contains the terms \( \frac{10}{x} \) and \( \frac{12}{x-3} \). A common denominator for these fractions is \( x(x-3) \). We will use this denominator to combine the fractions.
02
Rewrite Each Term with the Common Denominator
Multiply each side of the equation by \( x(x-3) \) to eliminate the denominators:\[(x(x-3)) \left( \frac{10}{x} \right) - (x(x-3)) \left( \frac{12}{x-3} \right) + 4(x(x-3)) = 0\]This simplifies to:\[10(x-3) - 12x + 4x(x-3) = 0\]
03
Simplify the Equation
Distribute and simplify the terms:\[10x - 30 - 12x + 4x^2 - 12x = 0\]Combine like terms:\[4x^2 - 14x - 30 = 0\]
04
Solve the Quadratic Equation
We have a quadratic equation \(4x^2 - 14x - 30 = 0\). Use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]where \(a = 4\), \(b = -14\), and \(c = -30\).
05
Calculate the Discriminant
Calculate the discriminant to check the nature of the roots:\[b^2 - 4ac = (-14)^2 - 4 \times 4 \times (-30) = 196 + 480 = 676\]Since the discriminant is positive, there are two real solutions.
06
Find the Roots Using the Quadratic Formula
Use the quadratic formula with the calculated values:\[x = \frac{14 \pm \sqrt{676}}{8}\]Simplifying:\[x = \frac{14 \pm 26}{8}\]Thus, the possible solutions are \(x = \frac{40}{8} = 5\) and \(x = \frac{-12}{8} = -1.5\).
07
Verify the Solutions
Check if these solutions satisfy the original equation and do not lead to undefined terms:- For \(x = 5\): \(\frac{10}{5} - \frac{12}{2} + 4 = 2 - 6 + 4 = 0\), valid.- For \(x = -1.5\): \(\frac{10}{-1.5} - \frac{12}{-4.5} + 4 = \approx -6.67 + 2.67 + 4 = 0\), valid.There are no values of \(x\) that lead to division by zero, so both are valid.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Common Denominator
When working with fractions in equations, a common denominator helps simplify the process by combining the fractions into a single expression. In this case, the equation is \[\frac{10}{x} - \frac{12}{x-3} + 4 = 0.\] To combine these fractions, we identify a common denominator.
- The terms \(\frac{10}{x}\) and \(\frac{12}{x-3}\) have denominators \(x\) and \(x-3\).
- The expression \(x(x-3)\) serves as the least common denominator (LCD).
Quadratic Equation
A quadratic equation is an equation of the form \[ax^2 + bx + c = 0,\] where \(a\), \(b\), and \(c\) are constants. These equations are fundamental in algebra and often arise when you simplify polynomial expressions.
- In our simplified problem, we ended up with the quadratic equation: \(4x^2 - 14x - 30 = 0\).
- This comes from combining and simplifying like terms of the expression prior multiplying by \(x(x-3)\).
Discriminant
The discriminant of a quadratic equation provides insight into the nature of its solutions, specifically if they are real or complex, and how many solutions exist. The discriminant is derived from the formula: \[b^2 - 4ac.\]
- If the discriminant is positive, there are two distinct real solutions.
- If it is zero, there is exactly one real solution (also called a repeated root).
- If the discriminant is negative, the equation has no real solutions but two complex solutions.
Quadratic Formula
The quadratic formula is a reliable method to find the solutions of any quadratic equation. The formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.\]This formula encompasses all cases that can arise with a quadratic equation by considering the factors \(a\), \(b\), and \(c\). In the given exercise,
- The values of \(a\), \(b\), and \(c\) are 4, -14, and -30 respectively.
- Substituting these into the quadratic formula gives \[x = \frac{14 \pm 26}{8},\] which simplifies to two potential solutions: \(x = 5\) and \(x = -1.5\).
