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Solve by completing the square.\newlinez2+6z=1z^2 + 6z = 1\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinez=z = _____ or z=z = _____

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Q. Solve by completing the square.\newlinez2+6z=1z^2 + 6z = 1\newlineWrite your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.\newlinez=z = _____ or z=z = _____
  1. Write Equation Form: Write the equation in the form of z2+bz=cz^2 + bz = c. The given equation is already in this form: z2+6z=1z^2 + 6z = 1.
  2. Move Constant Term: Move the constant term to the other side of the equation.\newlineSubtract 11 from both sides to isolate the zz terms:\newlinez2+6z−1=0z^2 + 6z - 1 = 0\newlinez2+6z=1z^2 + 6z = 1
  3. Find Completing Number: Find the number to complete the square.\newlineTo complete the square, we need to add (b/2)2(b/2)^2 to both sides of the equation, where bb is the coefficient of zz. In this case, b=6b = 6, so (6/2)2=32=9(6/2)^2 = 3^2 = 9.\newlineAdd 99 to both sides of the equation:\newlinez2+6z+9=1+9z^2 + 6z + 9 = 1 + 9\newlinez2+6z+9=10z^2 + 6z + 9 = 10
  4. Factor Left Side: Factor the left side of the equation.\newlineThe left side of the equation is now a perfect square trinomial:\newline(z+3)2=10(z + 3)^2 = 10
  5. Take Square Root: Take the square root of both sides of the equation.\newlineTo solve for zz, take the square root of both sides:\newline(z+3)2=±10\sqrt{(z + 3)^2} = \pm\sqrt{10}\newlinez+3=±10z + 3 = \pm\sqrt{10}
  6. Solve for z: Solve for z.\newlineSubtract 33 from both sides to isolate zz:\newlinez=−3±10z = -3 \pm \sqrt{10}
  7. Simplify Square Root: Simplify the square root and round to the nearest hundredth if necessary.\newline10\sqrt{10} is an irrational number, so we will round it to the nearest hundredth:\newline10≈3.16\sqrt{10} \approx 3.16\newlineTherefore, z≈−3±3.16z \approx -3 \pm 3.16
  8. Find Two Values: Find the two values of zz.z≈−3+3.16z \approx -3 + 3.16 implies z≈0.16z \approx 0.16z≈−3−3.16z \approx -3 - 3.16 implies z≈−6.16z \approx -6.16Values of zz: 0.160.16, −6.16-6.16

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