Question

rewrite the equation of the circle (x^{2}+y^{2}+10x - 4y+25 = 0) into standard form. enter your answers in the boxes to complete the equation. be sure to include an addition or subtraction symbol where needed. ((xsquare)^{2}+(ysquare)^{2}=square)

Explanation:

Step1: Group x - terms and y - terms

Group the x - terms and y - terms in the equation \(x^{2}+y^{2}+10x - 4y+25 = 0\): \((x^{2}+10x)+(y^{2}-4y)+25 = 0\).

Step2: Complete the square for x - terms

For the x - terms \(x^{2}+10x\), use the formula \((a + b)^2=a^{2}+2ab + b^{2}\). Here \(a=x\) and \(2b = 10\) (so \(b = 5\)), and \(x^{2}+10x=(x + 5)^{2}-25\).

Step3: Complete the square for y - terms

For the y - terms \(y^{2}-4y\), using the formula \((a + b)^2=a^{2}+2ab + b^{2}\) (or \((a - b)^2=a^{2}-2ab + b^{2}\)), here \(a = y\) and \(2b=4\) (so \(b = 2\)), and \(y^{2}-4y=(y - 2)^{2}-4\).

Step4: Substitute into the original equation

Substitute the completed - square expressions into the original equation: \((x + 5)^{2}-25+(y - 2)^{2}-4+25 = 0\).

Step5: Simplify the equation

Simplify the left - hand side of the equation: \((x + 5)^{2}+(y - 2)^{2}=4\).

Answer:

\((x + 5)^{2}+(y - 2)^{2}=4\)