Question

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≠,

Answer 1

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:

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