To find the zeros of the quadratic function \( y = 2x^2 + 5x - 12 \), we need to solve the equation \( 2x^2 + 5x - 12 = 0 \).
We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \( a = 2 \), \( b = 5 \), and \( c = -12 \).
- Calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 5^2 - 4 \cdot 2 \cdot (-12) = 25 + 96 = 121 \]
- Substitute \( a \), \( b \), and the discriminant into the quadratic formula:
\[ x = \frac{-5 \pm \sqrt{121}}{2 \cdot 2} \]
\[ x = \frac{-5 \pm 11}{4} \]
- Calculate the two potential solutions:
- For the positive case:
\[ x = \frac{-5 + 11}{4} = \frac{6}{4} = \frac{3}{2} \]
- For the negative case:
\[ x = \frac{-5 - 11}{4} = \frac{-16}{4} = -4 \]
Thus, the zeros of the graph are:
\[ x = \frac{3}{2} \quad \text{and} \quad x = -4 \]
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