To solve the equation \( x^2 - 6x = 10 \) using the quadratic formula, follow these steps in order:
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Rearrange the equation to standard form: Move all terms to one side of the equation so that it is in the form \( ax^2 + bx + c = 0 \). \[ x^2 - 6x - 10 = 0 \]
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Identify coefficients: From the standard form \( ax^2 + bx + c = 0 \), identify the coefficients:
- \( a = 1 \)
- \( b = -6 \)
- \( c = -10 \)
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Use the quadratic formula: The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute \( a \), \( b \), and \( c \) into the formula.
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Calculate the discriminant: Calculate \( b^2 - 4ac \): \[ (-6)^2 - 4(1)(-10) = 36 + 40 = 76 \]
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Calculate the roots using the square root of the discriminant: Now substitute back into the quadratic formula: \[ x = \frac{-(-6) \pm \sqrt{76}}{2 \cdot 1} \] Simplifying further: \[ x = \frac{6 \pm \sqrt{76}}{2} \]
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Simplify further if necessary: Since \( \sqrt{76} \) can be simplified to \( 2\sqrt{19} \): \[ x = \frac{6 \pm 2\sqrt{19}}{2} \] Simplifying gives: \[ x = 3 \pm \sqrt{19} \]
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Write down the final answers: The solutions are: \[ x = 3 + \sqrt{19} \quad \text{and} \quad x = 3 - \sqrt{19} \]
So the correct order of steps to solve the equation \( x^2 - 6x = 10 \) using the quadratic formula is as follows:
- Rearrange the equation to standard form.
- Identify coefficients.
- Use the quadratic formula.
- Calculate the discriminant.
- Calculate the roots using the square root of the discriminant.
- Simplify further if necessary.
- Write down the final answers.