To solve the quadratic equation \( 2x^2 + 7x - 15 = 0 \) using the quadratic formula, we can start by identifying the coefficients \( a \), \( b \), and \( c \):
- \( a = 2 \)
- \( b = 7 \)
- \( c = -15 \)
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Now, we can calculate the discriminant (\( b^2 - 4ac \)):
\[ b^2 = 7^2 = 49 \] \[ 4ac = 4 \times 2 \times (-15) = -120 \] \[ b^2 - 4ac = 49 - (-120) = 49 + 120 = 169 \]
Now that we have the discriminant, we can substitute these values into the quadratic formula:
\[ x = \frac{-7 \pm \sqrt{169}}{2 \times 2} \]
Calculating the square root and simplifying:
\[ \sqrt{169} = 13 \]
So, we can now substitute that in:
\[ x = \frac{-7 \pm 13}{4} \]
This gives us two possible solutions:
- \( x = \frac{-7 + 13}{4} = \frac{6}{4} = \frac{3}{2} \)
- \( x = \frac{-7 - 13}{4} = \frac{-20}{4} = -5 \)
Therefore, the solutions to the quadratic equation \( 2x^2 + 7x - 15 = 0 \) are:
\[ x = \frac{3}{2} \quad \text{and} \quad x = -5 \]
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