To find the value of \( x \) that solves the equation \( 8x^2 - 2x - 3 = 0 \), we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this case, the coefficients are:
- \( a = 8 \)
- \( b = -2 \)
- \( c = -3 \)
Now, we will substitute these values into the quadratic formula.
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Calculate the discriminant \( b^2 - 4ac \): \[ b^2 = (-2)^2 = 4 \] \[ 4ac = 4 \cdot 8 \cdot (-3) = -96 \] \[ b^2 - 4ac = 4 - (-96) = 4 + 96 = 100 \]
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Now plug the values into the quadratic formula: \[ x = \frac{-(-2) \pm \sqrt{100}}{2 \cdot 8} \] \[ x = \frac{2 \pm 10}{16} \]
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Now calculate the two potential solutions:
- For the plus case: \[ x = \frac{2 + 10}{16} = \frac{12}{16} = \frac{3}{4} \]
- For the minus case: \[ x = \frac{2 - 10}{16} = \frac{-8}{16} = -\frac{1}{2} \]
Thus, the solutions to the equation \( 8x^2 - 2x - 3 = 0 \) are \( x = \frac{3}{4} \) and \( x = -\frac{1}{2} \).
Both values are solutions.