To rewrite the given equation \(3 + g^2 - 8g + 5g = 56g + 12\) as two equations joined by "and," we first simplify the original equation.
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Combine like terms on the left side: \[ 3 + g^2 - 8g + 5g = g^2 - 3g + 3 \] So, the equation becomes: \[ g^2 - 3g + 3 = 56g + 12 \]
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Move all terms to one side of the equation: \[ g^2 - 3g + 3 - 56g - 12 = 0 \] Combine like terms: \[ g^2 - 59g - 9 = 0 \]
Now we can express this as the two separate equations:
\[ g^2 - 59g - 9 = 0 \quad \text{and} \quad g \neq \text{(the roots of the equation, if necessary)} \]
To find the roots of the quadratic equation \(g^2 - 59g - 9 = 0\), you can use the quadratic formula \(g = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
Substituting \(a = 1\), \(b = -59\), and \(c = -9\):
\[ g = \frac{59 \pm \sqrt{(-59)^2 - 4(1)(-9)}}{2(1)} \] \[ g = \frac{59 \pm \sqrt{3481 + 36}}{2} \] \[ g = \frac{59 \pm \sqrt{3517}}{2} \]
Approximating \(\sqrt{3517} \approx 59.3\) (for rough calculation, better to leave it in exact form if fractions are needed),
the roots can be given as: \[ g_1 \approx \frac{59 + 59.3}{2} \quad \text{and} \quad g_2 \approx \frac{59 - 59.3}{2} \] \[ g_1 \approx 59.15 \quad \text{and} \quad g_2 \approx -0.15 \]
Thus, the rewritten statement could be: \[ g^2 - 59g - 9 = 0 \quad \text{and} \quad g \neq \frac{59 \pm \sqrt{3517}}{2} \]
If you were looking for a numerical answer, using the roots on the left and stating that \(g\) cannot equal these exact values would give the final format as required. Just specify the approximations in decimal or fractional answer as needed.