Write the equation of the circle in Standard Form. 8x+x^2-2y=64-y^2

## Step 1<br />The given equation is \(8x + x^{2} - 2y = 64 - y^{2}\). We need to rearrange this equation to the standard form of a circle equation, which is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.<br /><br />## Step 2<br />First, we rearrange the equation to group the \(x\) terms and \(y\) terms together. This gives us \(x^{2} + 8x - y^{2} + 2y = 64\).<br /><br />## Step 3<br />Next, we complete the square for the \(x\) terms and \(y\) terms. This involves adding and subtracting the square of half the coefficient of \(x\) and \(y\) respectively.<br /><br />### For \(x\), we add and subtract \((\frac{8}{2})^2 = 16\).<br />### For \(y\), we add and subtract \((\frac{2}{2})^2 = 1\).<br /><br />## Step 4<br />Adding these to the equation gives us \(x^{2} + 8x + 16 - y^{2} + 2y + 1 = 64 + 16 + 1\).<br /><br />## Step 5<br />Simplifying this gives us \((x + 4)^2 - (y - 1)^2 = 81\).<br /><br />## Step 6<br />Finally, we rearrange the equation to the standard form of a circle equation, which gives us \((x + 4)^2 + (y - 1)^2 = 81\).