Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![**Problem Statement:**
Given the function \( f(x) = 3x^2 + 6x - 4 \), find the values of \( x \) such that \( f(x) = 0 \) using an algebraic method. Express answers in simplest radical form.
**Steps to Solve:**
1. **Identify the given quadratic equation:**
\[
f(x) = 3x^2 + 6x - 4
\]
2. **Set the equation to zero for solving:**
\[
3x^2 + 6x - 4 = 0
\]
3. **Use the quadratic formula, where \( a = 3 \), \( b = 6 \), and \( c = -4 \):**
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
4. **Calculate the discriminant:**
\[
b^2 - 4ac = 6^2 - 4 \times 3 \times (-4)
\]
\[
= 36 + 48 = 84
\]
5. **Substitute into the quadratic formula:**
\[
x = \frac{-6 \pm \sqrt{84}}{6}
\]
6. **Simplify the square root and expression:**
\[
\sqrt{84} = \sqrt{4 \times 21} = 2\sqrt{21}
\]
\[
x = \frac{-6 \pm 2\sqrt{21}}{6}
\]
7. **Simplify further:**
\[
x = \frac{-3 \pm \sqrt{21}}{3}
\]
Therefore, the solutions in simplest radical form are:
\[
x = \frac{-3 + \sqrt{21}}{3}, \quad x = \frac{-3 - \sqrt{21}}{3}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4cc4969d-0d82-4159-a2f2-9f3983870385%2Ff922192a-8c8d-4da3-ae00-4aa83bc2f2e8%2Fkihv5toj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given the function \( f(x) = 3x^2 + 6x - 4 \), find the values of \( x \) such that \( f(x) = 0 \) using an algebraic method. Express answers in simplest radical form.
**Steps to Solve:**
1. **Identify the given quadratic equation:**
\[
f(x) = 3x^2 + 6x - 4
\]
2. **Set the equation to zero for solving:**
\[
3x^2 + 6x - 4 = 0
\]
3. **Use the quadratic formula, where \( a = 3 \), \( b = 6 \), and \( c = -4 \):**
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
4. **Calculate the discriminant:**
\[
b^2 - 4ac = 6^2 - 4 \times 3 \times (-4)
\]
\[
= 36 + 48 = 84
\]
5. **Substitute into the quadratic formula:**
\[
x = \frac{-6 \pm \sqrt{84}}{6}
\]
6. **Simplify the square root and expression:**
\[
\sqrt{84} = \sqrt{4 \times 21} = 2\sqrt{21}
\]
\[
x = \frac{-6 \pm 2\sqrt{21}}{6}
\]
7. **Simplify further:**
\[
x = \frac{-3 \pm \sqrt{21}}{3}
\]
Therefore, the solutions in simplest radical form are:
\[
x = \frac{-3 + \sqrt{21}}{3}, \quad x = \frac{-3 - \sqrt{21}}{3}
\]
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