Question

solve the quadratic equation by completing the square. (x^{2}+6x - 1 = 0). first, choose the appropriate form and fill in the blanks with the correct numbers. then, solve the equation. round your answer to the nearest hundredth. if there is more than one solution, separate them with commas. form: ((x+square)^{2}=square) ((x - square)^{2}=square) solution: (x=square)

Explanation:

Step1: Rewrite the equation for completing square

For the quadratic equation \(x^{2}+6x - 1=0\), we know that \((a + b)^2=a^{2}+2ab + b^{2}\). In \(x^{2}+6x\), if \(a = x\) and \(2ab=6x\), then \(b = 3\). So we rewrite the equation as \(x^{2}+6x=1\), and then \(x^{2}+6x + 9=1 + 9\).
\[x^{2}+6x+9 = 10\]
\[(x + 3)^{2}=10\]

Step2: Solve for x

Take the square - root of both sides:
\[x+3=\pm\sqrt{10}\]
\[x=-3\pm\sqrt{10}\]
\[x=-3+\sqrt{10}\approx - 3 + 3.162 = 0.16\]
\[x=-3-\sqrt{10}\approx-3 - 3.162=-6.16\]

Answer:

\(x=-3+\sqrt{10}\approx0.16,x=-3 - \sqrt{10}\approx - 6.16\); \((x + 3)^{2}=10\)