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Mobile App Chapter 5: Problem 16
For which values of \(x\) is \(-x^2+4 x-3\) positive?
A. \(1>x>3\)
B. \(x<-3\)
C. \(-4
Short Answer
Expert verified
D. \(1 < x < 3\)
Step by step solution
01
Write the Inequality
We need to find the values of \(x\) for which the quadratic expression \(-x^2 + 4x - 3\) is positive. Thus, we set up the inequality: \(-x^2 + 4x - 3 > 0\)
02
Find the Roots of the Quadratic Equation
First, solve the corresponding quadratic equation \(-x^2 + 4x - 3 = 0\). We use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, \(a = -1\), \(b = 4\), and \(c = -3\). Substituting these values in, we get: \[ x = \frac{-4 \pm \sqrt{16 - 12}}{-2} \]\[ x = \frac{-4 \pm 2}{-2} \]Hence, \(x = 1\) and \(x = 3\).
03
Sign Analysis of the Quadratic Expression
We next analyze the sign of the quadratic expression \(-x^2 + 4x - 3\) between the roots. These intervals are \((-\infty, 1)\), \((1, 3)\), and \((3, \infty)\).- For \(x < 1\) (e.g., \(x = 0\)), \(-x^2 + 4x - 3\) becomes \(0 + 0 - 3 = -3\). It is negative.- For \(1 < x < 3\) (e.g., \(x = 2\)), \(-x^2 + 4x - 3\) becomes \(-4 + 8 - 3 = 1\). It is positive.- For \(x > 3\) (e.g., \(x = 4\)), \(-x^2 + 4x - 3\) becomes \(-16 + 16 - 3 = -3\). It is negative.
04
Determine the Correct Interval
Based on the sign analysis, the quadratic expression \(-x^2 + 4x - 3\) is positive only for the interval \(1 < x < 3\). Thus, the correct answer is option D - \(1 < x < 3\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
GMAT Preparation
Preparing for the GMAT can be a daunting task, but understanding fundamental math concepts like quadratic inequalities is key. The GMAT often tests your ability to analyze and solve these types of problems efficiently.
Make sure you are comfortable with:
Make sure you are comfortable with:
- Identifying the standard form of quadratic equations.
- Finding the roots using the quadratic formula.
- Performing a sign analysis to find where the quadratic expression is positive or negative.
Quadratic Equations
A quadratic equation is any equation that can be written in the form ax^2 + bx + c = 0. Identify the coefficients (a, b, c), then use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).In the given example, \(a = -1\), \(b = 4\), and \(c = -3\).
After solving, you'll find the roots to be \(x = 1\) and \(x = 3\). These roots are essential for the next steps, such as sign analysis. Quadratic equations often serve as the foundation for more complex problems, making them crucial for any GMAT preparation.
After solving, you'll find the roots to be \(x = 1\) and \(x = 3\). These roots are essential for the next steps, such as sign analysis. Quadratic equations often serve as the foundation for more complex problems, making them crucial for any GMAT preparation.
Inequality Analysis
Inequalities involve finding the range of values where an expression holds true. For quadratic inequalities, follow these steps:
In this example, analyze the intervals (\((-\infty, 1)\), \((1, 3)\), \((3, \infty)\)). Only the interval (\(1 < x < 3)\) makes \(-x^2 + 4x - 3\) positive, so that's the solution.
- Write down the inequality and turn it into an equation to find the roots.
- Use these roots to divide the number line into intervals.
- Choose a test point from each interval and plug it back into the original inequality to see if it makes the expression positive or negative.
In this example, analyze the intervals (\((-\infty, 1)\), \((1, 3)\), \((3, \infty)\)). Only the interval (\(1 < x < 3)\) makes \(-x^2 + 4x - 3\) positive, so that's the solution.
Mathematical Problem Solving
Effective mathematical problem solving involves breaking a problem into manageable steps. Here's a quick guide using the given quadratic inequality:
1. **Understand the Problem**: We need to find when \(-x^2 + 4x - 3\) is positive.
2. **Formulate the Plan**: Convert the inequality to an equation to find the roots using the quadratic formula.
3. **Execute the Plan**: Solve for the roots (\(x = 1\) and \(x = 3\)).
4. **Check the Intervals**: Perform a sign analysis between the intervals (\(1, 3)\). This method ensures you cover all possibilities systematically, leading to the final solution (1 < x < 3).
Enhancing these problem-solving skills is essential for excelling in standardized tests like the GMAT.
1. **Understand the Problem**: We need to find when \(-x^2 + 4x - 3\) is positive.
2. **Formulate the Plan**: Convert the inequality to an equation to find the roots using the quadratic formula.
3. **Execute the Plan**: Solve for the roots (\(x = 1\) and \(x = 3\)).
4. **Check the Intervals**: Perform a sign analysis between the intervals (\(1, 3)\). This method ensures you cover all possibilities systematically, leading to the final solution (1 < x < 3).
Enhancing these problem-solving skills is essential for excelling in standardized tests like the GMAT.
