To solve the quadratic equation \( y^2 - 12y + 11 = 0 \), we can use the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this equation, \( a = 1 \), \( b = -12 \), and \( c = 11 \).
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Calculate the discriminant (\( b^2 - 4ac \)): \[ b^2 = (-12)^2 = 144 \] \[ 4ac = 4 \cdot 1 \cdot 11 = 44 \] \[ b^2 - 4ac = 144 - 44 = 100 \]
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Now, plug the values into the quadratic formula: \[ y = \frac{-(-12) \pm \sqrt{100}}{2 \cdot 1} \] \[ y = \frac{12 \pm 10}{2} \]
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This gives us two possible solutions: \[ y = \frac{12 + 10}{2} = \frac{22}{2} = 11 \] \[ y = \frac{12 - 10}{2} = \frac{2}{2} = 1 \]
Thus, the solutions to the equation \( y^2 - 12y + 11 = 0 \) are \( y = 11 \) and \( y = 1 \).