Question

Use a graphing utility as an aid in factoring to solve the equation.
3xsquaredplus14xminus49equals0
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your answer.
A.
The​ solution(s) is/are
  
enter your response here.
​(Type an integer or a simplified fraction. Use a comma to separate answers as​ needed.)
B.
The equation has no solution.

Answer 1

To solve the equation \(3x^2 + 14x - 49 = 0\), we can start by identifying the coefficients and then use the quadratic formula or a graphing utility to solve for \(x\).

The coefficients are:
- \(a = 3\)
- \(b = 14\)
- \(c = -49\)

### Step 1: Finding the Discriminant
The discriminant \((D)\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by:
\[ D = b^2 - 4ac \]

Calculating the discriminant:
\[ D = 14^2 - 4(3)(-49) \]
\[ D = 196 + 588 \]
\[ D = 784 \]

Since the discriminant is positive, this means there are two real and distinct solutions.

### Step 2: Using the Quadratic Formula
The solutions can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]

Substituting the values:
\[ x = \frac{-14 \pm \sqrt{784}}{2 \cdot 3} \]
\[ x = \frac{-14 \pm 28}{6} \]

### Step 3: Calculating the Two Solutions
1. **First Solution:**
\[ x = \frac{-14 + 28}{6} = \frac{14}{6} = \frac{7}{3} \]

2. **Second Solution:**
\[ x = \frac{-14 - 28}{6} = \frac{-42}{6} = -7 \]

### Final Answers
The solutions to the equation \(3x^2 + 14x - 49 = 0\) are:
- \(x = \frac{7}{3}\)
- \(x = -7\)

Thus, the answer is:
**A. The solutions are \(\frac{7}{3}, -7\).**