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Solve by completing the square.\newlinez2+8z=21z^2 + 8z = 21\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+8z=21z^2 + 8z = 21\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. Rewrite equation as z2+bz=cz^2 + bz = c: Rewrite the equation in the form of z2+bz=cz^2 + bz = c.\newlineSubtract 2121 from both sides to set the equation to zero.\newlinez2+8z−21=0z^2 + 8z - 21 = 0
  2. Move constant term to right: Move the constant term to the right side of the equation. z2+8z=21z^2 + 8z = 21
  3. Complete the square: Complete the square by adding the square of half the coefficient of zz to both sides.\newlineSince (82)2=16(\frac{8}{2})^2 = 16, add 1616 to both sides.\newlinez2+8z+16=21+16z^2 + 8z + 16 = 21 + 16\newlinez2+8z+16=37z^2 + 8z + 16 = 37
  4. Factor left side: Factor the left side of the equation.\newline(z+4)2=37(z + 4)^2 = 37
  5. Take square root: Take the square root of both sides of the equation.\newline(z+4)2=±37\sqrt{(z + 4)^2} = \pm\sqrt{37}\newlinez+4=±37z + 4 = \pm\sqrt{37}
  6. Solve for z: Solve for z by isolating the variable.\newlineSubtract 44 from both sides of the equation.\newlinez+4−4=±37−4z + 4 - 4 = \pm\sqrt{37} - 4\newlinez=−4±37z = -4 \pm \sqrt{37}
  7. Calculate decimal values: Calculate the approximate decimal values of zz, rounded to the nearest hundredth.z≈−4+37z \approx -4 + \sqrt{37} implies z≈−4+6.08z \approx -4 + 6.08 which is z≈2.08z \approx 2.08.z≈−4−37z \approx -4 - \sqrt{37} implies z≈−4−6.08z \approx -4 - 6.08 which is z≈−10.08z \approx -10.08.

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