Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 4: Problem 30
For Problems \(1-44\), solve each equation. $$ \frac{x}{-4}=\frac{3}{12 x-25} $$
Short Answer
Expert verified
The solutions are \(x = \frac{4}{3}\) and \(x = \frac{3}{4}\).
Step by step solution
01
Understand the Problem
We need to solve the equation \( \frac{x}{-4}=\frac{3}{12x-25} \). The goal is to find the value of \(x\) that makes this equation true.
02
Cross Multiply
In the equation \( \frac{x}{-4} = \frac{3}{12x-25} \), cross-multiply to eliminate the fractions: \(x(12x-25) = 3(-4)\). This becomes \(12x^2 - 25x = -12\).
03
Rearrange the Equation
Move all terms to one side of the equation to make it a standard quadratic equation: \(12x^2 - 25x + 12 = 0\).
04
Apply the Quadratic Formula
For the quadratic equation \(ax^2 + bx + c = 0\), use the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 12\), \(b = -25\), and \(c = 12\).
05
Calculate the Discriminant
Compute the discriminant \(b^2 - 4ac\): \((-25)^2 - 4(12)(12) = 625 - 576 = 49\).
06
Solve for x using the Quadratic Formula
Since the discriminant is 49, we have two real solutions: \(x = \frac{-(-25) \pm \sqrt{49}}{2(12)}\). Solve for both solutions, \(x = \frac{25 \pm 7}{24}\).
07
Find the Two Possible Solutions
Compute both solutions: \(x = \frac{25 + 7}{24} = \frac{32}{24} = \frac{4}{3}\) and \(x = \frac{25 - 7}{24} = \frac{18}{24} = \frac{3}{4}\).
08
Verify the Solutions
Plug \(x = \frac{4}{3}\) and \(x = \frac{3}{4}\) back into the original equation to ensure they satisfy it. Both values make the equation true, confirming they are correct solutions.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cross Multiplication
Cross multiplication is a technique used to solve equations that involve two fractions set equal to each other. This method helps to eliminate the fractions, making the equation easier to solve.
- Cross-multiplication works by multiplying across the equation diagonally. In other words, you multiply the numerator of one fraction by the denominator of the other fraction.
- For our problem, we have \( \frac{x}{-4} = \frac{3}{12x-25} \). To cross-multiply, multiply \(x\) by \(12x - 25\) and \(-4\) by \(3\).
- This results in the equation \(x(12x-25) = 3(-4)\), which simplifies to \(12x^2 - 25x = -12\).
Quadratic Formula
The quadratic formula is a powerful tool for finding the solutions to any quadratic equation of the form \(ax^2 + bx + c = 0\). This formula is especially useful when the equation cannot be easily factored. The quadratic formula is given as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- In our equation, \(12x^2 - 25x + 12 = 0\), we identify \(a = 12\), \(b = -25\), and \(c = 12\).
- Substitute these values into the formula to solve for \(x\).
- Using the formula, substitute to find \(x = \frac{-(-25) \pm \sqrt{49}}{24}\).
Discriminant Calculation
The discriminant is a part of the quadratic formula that allows us to determine the number and type of solutions for a quadratic equation. It is calculated from the expression \(b^2 - 4ac\).
- The discriminant is crucial as it tells us whether the solutions to the quadratic equation are real or complex.
- In this problem, we calculate the discriminant as \((-25)^2 - 4(12)(12) = 625 - 576 = 49\).
- Positive, as in our case (49), there are two distinct real solutions.
- Zero, there is exactly one real solution, meaning it's a perfect square trinomial.
- Negative, there are no real solutions; instead, the solutions are complex numbers.
