Question
Solve the equation \( 6 x^{2}+13 x+6=0 \) Answer: \( x=\square \)
Ask by Rodriguez Ford. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( x = -\frac{3}{2} \) or \( x = -\frac{2}{3} \)
Solution
1. **Write the equation:**
\[
6x^2+13x+6=0
\]
2. **Factor the quadratic equation:**
We look for two numbers that multiply to \(6 \times 6 = 36\) and add to \(13\). The numbers \(9\) and \(4\) satisfy these conditions because \(9 \times 4 = 36\) and \(9 + 4 = 13\).
3. **Rewrite the middle term using \(9x\) and \(4x\):**
\[
6x^2+9x+4x+6=0
\]
4. **Factor by grouping:**
Group the terms:
\[
(6x^2+9x)+(4x+6)=0
\]
Factor out the common factors in each group:
- From the first group \((6x^2+9x)\), factor out \(3x\):
\[
3x(2x+3)
\]
- From the second group \((4x+6)\), factor out \(2\):
\[
2(2x+3)
\]
The equation becomes:
\[
3x(2x+3)+2(2x+3)=0
\]
5. **Factor out the common factor \((2x+3)\):**
\[
(2x+3)(3x+2)=0
\]
6. **Solve for \(x\) by setting each factor equal to 0:**
- Set \(2x+3=0\):
\[
2x=-3 \quad \Rightarrow \quad x=-\frac{3}{2}
\]
- Set \(3x+2=0\):
\[
3x=-2 \quad \Rightarrow \quad x=-\frac{2}{3}
\]
7. **Final Answer:**
\[
x=-\frac{3}{2} \quad \text{or} \quad x=-\frac{2}{3}
\]
Answered by UpStudy AI and reviewed by a Professional Tutor
Like
error msg
Explain 
Simplify this solution The Deep Dive
To solve the quadratic equation \( 6 x^{2}+13 x+6=0 \), we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 6 \), \( b = 13 \), and \( c = 6 \). First, calculate the discriminant: \[ b^2 - 4ac = 13^2 - 4 \cdot 6 \cdot 6 = 169 - 144 = 25. \] Now, applying the quadratic formula: \[ x = \frac{-13 \pm \sqrt{25}}{2 \cdot 6} = \frac{-13 \pm 5}{12}. \] This gives us two solutions: 1. \( x = \frac{-13 + 5}{12} = \frac{-8}{12} = -\frac{2}{3} \) 2. \( x = \frac{-13 - 5}{12} = \frac{-18}{12} = -\frac{3}{2} \) So the solutions to the equation are: \[ x = -\frac{2}{3}, -\frac{3}{2}. \] Answer: \( x = -\frac{2}{3}, -\frac{3}{2} \)
