Questions: Complete the square and write the given equation in standard form. Then give the center and radius of the circle and graph the equation. x^2+y^2+8 x-4 y-5=0 The equation of the circle in standard form is (Simplify your answer.)

Solution Steps

The given equation is: \[ x^{2}+y^{2}+8x-4y-5=0 \]

Group the \(x\) and \(y\) terms together: \[ (x^{2} + 8x) + (y^{2} - 4y) = 5 \]

Complete the square for the \(x\) terms: \[ x^{2} + 8x = (x + 4)^{2} - 16 \]

Complete the square for the \(y\) terms: \[ y^{2} - 4y = (y - 2)^{2} - 4 \]

Substitute the completed squares back into the equation: \[ (x + 4)^{2} - 16 + (y - 2)^{2} - 4 = 5 \]

Simplify the equation: \[ (x + 4)^{2} + (y - 2)^{2} - 20 = 5 \] \[ (x + 4)^{2} + (y - 2)^{2} = 25 \]

The equation of the circle in standard form is: \[ (x + 4)^{2} + (y - 2)^{2} = 25 \] The center of the circle is \((-4, 2)\) and the radius is \(5\).

Final Answer