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
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First Solution: \[ x = \frac{-14 + 28}{6} = \frac{14}{6} = \frac{7}{3} \]
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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 \(\frac{7}{3}, -7\).