Rewrite the equation 3+g2−8g+5g=56g+12as two equations joined by “and.” Enter the smaller number first and use fractions if needed.(1 point)3+g2−8g+5g=56g+12and g≠,

1 answer

To rewrite the equation \(3 + g^2 - 8g + 5g = 56g + 12\) as two equations joined by "and," we first simplify the equation.

Combine the terms on the left:

\[ g^2 - 3g + 3 = 56g + 12 \]

Then, move all terms to one side to set the equation to 0:

\[ g^2 - 3g - 56g + 3 - 12 = 0 \]

Simplifying the equation, we get:

\[ g^2 - 59g - 9 = 0 \]

The two equations can be expressed as follows:

\[ g^2 - 59g - 9 = 0 \quad \text{and} \quad g \neq \text{something} \]

Next, we need to find a value for \(g\) such that the equation equals zero. We'll use the quadratic formula:

\[ g = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Where \(a = 1\), \(b = -59\), and \(c = -9\):

\[ g = \frac{59 \pm \sqrt{(-59)^2 - 4 \cdot 1 \cdot (-9)}}{2 \cdot 1} \] \[ g = \frac{59 \pm \sqrt{3481 + 36}}{2} \] \[ g = \frac{59 \pm \sqrt{3517}}{2} \]

Now, we can represent the two solutions approximately, but first we need the exact form. \(g\) is either a positive or negative solution, and for the inequality, we require \(g\) not to equal one of these roots.

Therefore, the equations are:

\[ g^2 - 59g - 9 = 0 \quad \text{and} \quad g \neq \frac{59 \pm \sqrt{3517}}{2} \]

We can approximate the roots as:

Calculating \( \sqrt{3517} \approx 59.3\), which gives us roots:

\[ g \approx \frac{59 + 59.3}{2} \quad \text{and} \quad g \approx \frac{59 - 59.3}{2} \]

This leads to:

\[ g \approx 59.15 \quad \text{and} \quad g \approx -0.15 \]

So, the final expression with approximate values can be:

\[ g^2 - 59g - 9 = 0 \quad \text{and} \quad g \neq -0.15 \]

Thus the equations would be:

\[ g^2 - 59g - 9 = 0 \quad \text{and} \quad g \neq -\frac{1}{6} \]

Final answer:

\[ g^2 - 59g - 9 = 0 \quad \text{and} \quad g \neq -\frac{1}{6} \]