To solve the equation \(x^2 - 10x = 12\) using the quadratic formula, we first need to rearrange the equation into standard form \(ax^2 + bx + c = 0\). Here are the steps in the correct order:
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Rearrange the equation: Move all terms to one side to set the equation to zero: \[ x^2 - 10x - 12 = 0 \]
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Identify coefficients: Identify the coefficients \(a\), \(b\), and \(c\) from the equation \(ax^2 + bx + c = 0\):
- \(a = 1\)
- \(b = -10\)
- \(c = -12\)
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Use the quadratic formula: Recall the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
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Calculate the discriminant: Compute the value of \(b^2 - 4ac\): \[ b^2 - 4ac = (-10)^2 - 4(1)(-12) = 100 + 48 = 148 \]
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Substitute into the formula: Plug the values of \(a\), \(b\), and the discriminant into the quadratic formula: \[ x = \frac{-(-10) \pm \sqrt{148}}{2(1)} = \frac{10 \pm \sqrt{148}}{2} \]
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Simplify the square root: Simplify \(\sqrt{148}\): \[ \sqrt{148} = \sqrt{4 \cdot 37} = 2\sqrt{37} \]
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Substitute back and simplify: \[ x = \frac{10 \pm 2\sqrt{37}}{2} = 5 \pm \sqrt{37} \]
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Write the final solutions: The final solutions are: \[ x = 5 + \sqrt{37} \quad \text{and} \quad x = 5 - \sqrt{37} \]
By following these steps in order, you can effectively solve the quadratic equation using the quadratic formula.