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Solve for b. sqrt(b+1)=sqrt(b+6)-1

Solve for bb.\newlineb+1=b+6−1 \sqrt{b+1}=\sqrt{b+6}-1

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Full solution

Q. Solve for bb.\newlineb+1=b+6−1 \sqrt{b+1}=\sqrt{b+6}-1
  1. Isolate bb in equation: We are given the equation b+1=b+6−1\sqrt{b+1} = \sqrt{b+6} - 1. To solve for bb, we need to isolate bb on one side of the equation.
  2. Square both sides: First, let's get rid of the square roots by squaring both sides of the equation. This will eliminate the square root on the left side and we will have to square the entire right side of the equation.\newline(\sqrt{b+\(1\)})^\(2 = (\sqrt{b+66} - 11)^22
  3. Apply binomial square formula: Squaring the left side gives us b+1b + 1. On the right side, we need to apply the binomial square formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.\newlineb+1=(b+6)2−2⋅b+6⋅1+12b + 1 = (\sqrt{b+6})^2 - 2\cdot\sqrt{b+6}\cdot 1 + 1^2
  4. Simplify right side: Squaring b+6\sqrt{b+6} gives us b+6b + 6. We also simplify the rest of the right side of the equation.b+1=b+6−2b+6+1b + 1 = b + 6 - 2\sqrt{b+6} + 1
  5. Combine like terms: Now, let's simplify the right side by combining like terms. b+1=b+7−2b+6b + 1 = b + 7 - 2\sqrt{b+6}
  6. Subtract bb from both sides: Next, we subtract bb from both sides to get the terms with bb on one side and the constant terms on the other side.b+1−b=b+7−b−2b+6b + 1 - b = b + 7 - b - 2\sqrt{b+6}
  7. Isolate term with square root: After simplifying, we have: \newline1=7−2b+61 = 7 - 2\sqrt{b+6}
  8. Divide both sides by −2-2: Now, we subtract 77 from both sides to isolate the term with the square root.\newline1−7=−2b+61 - 7 = -2\sqrt{b+6}
  9. Square both sides again: This simplifies to:\newline−6=−2b+6-6 = -2\sqrt{b+6}
  10. Solve for b: Next, we divide both sides by −2-2 to solve for b+6\sqrt{b+6}.−6/−2=−2⋅b+6/−2-6 / -2 = -2\cdot\sqrt{b+6} / -2
  11. Solve for b: Next, we divide both sides by −2-2 to solve for b+6\sqrt{b+6}.−6/−2=−2⋅b+6/−2-6 / -2 = -2\cdot\sqrt{b+6} / -2 This gives us:3=b+63 = \sqrt{b+6}
  12. Solve for b: Next, we divide both sides by −2-2 to solve for b+6\sqrt{b+6}.−6/−2=−2⋅b+6/−2-6 / -2 = -2\cdot\sqrt{b+6} / -2This gives us:3=b+63 = \sqrt{b+6}Now, we square both sides again to eliminate the square root and solve for b.32=(b+6)23^2 = (\sqrt{b+6})^2
  13. Solve for b: Next, we divide both sides by −2-2 to solve for b+6\sqrt{b+6}.−6/−2=−2⋅b+6/−2-6 / -2 = -2\cdot\sqrt{b+6} / -2 This gives us:3=b+63 = \sqrt{b+6} Now, we square both sides again to eliminate the square root and solve for bb.32=(b+6)23^2 = (\sqrt{b+6})^2 Squaring 33 gives us 99, and squaring b+6\sqrt{b+6} gives us b+6b + 6.b+6\sqrt{b+6}00
  14. Solve for b: Next, we divide both sides by −2-2 to solve for b+6\sqrt{b+6}.
    −6/−2=−2⋅b+6/−2-6 / -2 = -2\cdot\sqrt{b+6} / -2 This gives us:
    3=b+63 = \sqrt{b+6} Now, we square both sides again to eliminate the square root and solve for bb.
    32=(b+6)23^2 = (\sqrt{b+6})^2 Squaring 33 gives us 99, and squaring b+6\sqrt{b+6} gives us b+6b + 6.
    b+6\sqrt{b+6}00 Finally, we subtract b+6\sqrt{b+6}11 from both sides to solve for bb.
    b+6\sqrt{b+6}33
  15. Solve for b: Next, we divide both sides by −2-2 to solve for b+6\sqrt{b+6}.
    −6/−2=−2⋅b+6/−2-6 / -2 = -2\cdot\sqrt{b+6} / -2 This gives us:
    3=b+63 = \sqrt{b+6} Now, we square both sides again to eliminate the square root and solve for bb.
    32=(b+6)23^2 = (\sqrt{b+6})^2 Squaring 33 gives us 99, and squaring b+6\sqrt{b+6} gives us b+6b + 6.
    b+6\sqrt{b+6}00 Finally, we subtract b+6\sqrt{b+6}11 from both sides to solve for bb.
    b+6\sqrt{b+6}33 This gives us the final answer:
    b+6\sqrt{b+6}44

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