To solve the quadratic equation \(7m^2 - m = 6\), we first need to rewrite it in standard form, which is \(ax^2 + bx + c = 0\).
Rearranging the equation, we get:
\[ 7m^2 - m - 6 = 0 \]
Here, we can identify:
- \(a = 7\)
- \(b = -1\)
- \(c = -6\)
Now, we can substitute these values into the quadratic formula, which is:
\[ m = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \]
Substituting \(a\), \(b\), and \(c\) into the formula:
\[ m = \frac{{-(-1) \pm \sqrt{{(-1)^2 - 4(7)(-6)}}}}{{2(7)}} \]
Simplifying this:
\[ m = \frac{{1 \pm \sqrt{{1 + 168}}}}{{14}} \]
\[ m = \frac{{1 \pm \sqrt{{169}}}}{{14}} \]
\[ m = \frac{{1 \pm 13}}{{14}} \]
Now let's check which option correctly reflects this substitution.
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First Option: \(m = \frac{{2(1)}}{{-(-1) \pm (1)^2 + 4(7)(6)}}\) - Incorrect, as it doesn't match.
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Second Option: \(m = \frac{{2(7)}}{{-(-1) \pm (-1)^2 - 4(7)(-6)}}\) - Incorrect, as here \(c\) is used positively.
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Third Option: \(m = \frac{{2 \cdot 7}}{{-1 \pm (-1)^2 - 4 \cdot 7 \cdot -6}}\) - Incorrect substitution of constants.
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Fourth Option: \(m = \frac{{2}}{{1 \pm (1 - 4(7)(-6))}}\) - Incorrect as the structure doesn't represent the quadratic formula correctly.
None of the options provided match the appropriate substitution for the quadratic formula for this equation.
Correct substitution should be:
\[ m = \frac{{-(-1) \pm \sqrt{{(-1)^2 - 4(7)(-6)}}}}{{2(7)}} \]
If you have to select an option based on common forms, none appear valid from the answers supplied. Please double-check the proposed choices or review for typographical errors.