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Magazine Chapter 6: Problem 80
$$ \text { For Problems 51-80, solve each equation. (Objective 2) } $$ $$ 24 x^{2}+17 x-20=0 $$
Short Answer
Expert verified
The solutions are \( x = \frac{5}{8} \) and \( x = -\frac{4}{3} \).
Step by step solution
01
Identify the elements of the quadratic equation
The given equation is \( 24x^2 + 17x - 20 = 0 \). This is in the standard quadratic form: \( ax^2 + bx + c = 0 \). Here, \( a = 24 \), \( b = 17 \), and \( c = -20 \).
02
Use the quadratic formula
The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We will use it to find the roots of the given quadratic equation.
03
Calculate the discriminant
The discriminant is the part under the square root in the quadratic formula: \( b^2 - 4ac \). Calculate it using \( b = 17 \), \( a = 24 \), and \( c = -20 \): \( b^2 - 4ac = 17^2 - 4 \times 24 \times (-20) \).
04
Solve for the discriminant
Compute \( b^2 = 289 \) and \(-4ac = -4 \times 24 \times -20 = 1920 \), so the discriminant is \( 289 + 1920 = 2209 \).
05
Substitute values into the quadratic formula
Place the calculated values into the formula: \[ x = \frac{-17 \pm \sqrt{2209}}{48}. \]
06
Simplify the square root and solve for x
Since \( \sqrt{2209} = 47 \), the expressions become: \( x = \frac{-17 + 47}{48} \) and \( x = \frac{-17 - 47}{48} \).
07
Calculate each possible value for x
Calculating further gives \( x = \frac{30}{48} = \frac{5}{8} \) and \( x = \frac{-64}{48} = -\frac{4}{3} \).
08
Present the solution
The solutions to the equation \( 24x^2 + 17x - 20 = 0 \) are \( x = \frac{5}{8} \) and \( x = -\frac{4}{3} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a reliable tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). It is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].This formula provides a straightforward method to find the roots of any quadratic equation, regardless of whether the equation can be easily factored.
- **Step-by-Step Process:** - Identify coefficients \( a \), \( b \), and \( c \) from the quadratic equation. - Substitute these values into the quadratic formula. - Calculate the discriminant, \( b^2 - 4ac \), and use it to determine the number of real roots.
- **Benefits of Using the Quadratic Formula:** - Universally applicable: It works for all quadratic equations. - Simplifies complex problems: Especially when factoring is not easy.
- **Step-by-Step Process:** - Identify coefficients \( a \), \( b \), and \( c \) from the quadratic equation. - Substitute these values into the quadratic formula. - Calculate the discriminant, \( b^2 - 4ac \), and use it to determine the number of real roots.
- **Benefits of Using the Quadratic Formula:** - Universally applicable: It works for all quadratic equations. - Simplifies complex problems: Especially when factoring is not easy.
Discriminant
The discriminant is a fundamental part of the quadratic formula and plays a crucial role in determining the nature of a quadratic equation's roots. It is represented as \( D = b^2 - 4ac \).
- **Understanding the Discriminant's Role:** - It decides whether the roots are real or complex. - The value of the discriminant influenced how many solutions the quadratic equation has.
- **Interpretations of the Discriminant Value:** - **If** \( D > 0 \): - There are two distinct real roots. - Perfect square discriminant gives rational roots. - **If** \( D = 0 \): - There is exactly one real root (or a repeated root). - **If** \( D < 0 \): - The roots are complex and non-real.
Understanding and calculating the discriminant can vividly guide you through predicting the type of solutions.
- **Understanding the Discriminant's Role:** - It decides whether the roots are real or complex. - The value of the discriminant influenced how many solutions the quadratic equation has.
- **Interpretations of the Discriminant Value:** - **If** \( D > 0 \): - There are two distinct real roots. - Perfect square discriminant gives rational roots. - **If** \( D = 0 \): - There is exactly one real root (or a repeated root). - **If** \( D < 0 \): - The roots are complex and non-real.
Understanding and calculating the discriminant can vividly guide you through predicting the type of solutions.
Solving Equations
When solving quadratic equations, the aim is to find the value(s) of \( x \) that satisfy the equation. Quadratic equations can be solved using different methods, but one of the most robust is using the quadratic formula.
- **Steps in Solving Quadratic Equations:** - Begin by writing the equation in standard form \( ax^2 + bx + c = 0 \). - Calculate the discriminant \( D = b^2 - 4ac \) to understand the nature of the roots. - Use \( x = \frac{-b \pm \sqrt{D}}{2a} \) to find the roots.
- **Additional Considerations:** - **Factoring:** When applicable, factoring is quick, but not always feasible. - **Completing the Square:** Another technique that involves expressing the equation in a perfect square form. - **Graphing:** Visual representation can help in understanding the solutions better.
By methodically applying the quadratic formula and making sense of the discriminant, one can effectively solve quadratic equations across varying contexts.
- **Steps in Solving Quadratic Equations:** - Begin by writing the equation in standard form \( ax^2 + bx + c = 0 \). - Calculate the discriminant \( D = b^2 - 4ac \) to understand the nature of the roots. - Use \( x = \frac{-b \pm \sqrt{D}}{2a} \) to find the roots.
- **Additional Considerations:** - **Factoring:** When applicable, factoring is quick, but not always feasible. - **Completing the Square:** Another technique that involves expressing the equation in a perfect square form. - **Graphing:** Visual representation can help in understanding the solutions better.
By methodically applying the quadratic formula and making sense of the discriminant, one can effectively solve quadratic equations across varying contexts.
