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x2−10x+14=0x^{2}-10x+14=0\newlineOne solution to the given equation can be written as x=5+nx=5+\sqrt{n}, where nn is a constant. What is the value of nn?

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Q. x2−10x+14=0x^{2}-10x+14=0\newlineOne solution to the given equation can be written as x=5+nx=5+\sqrt{n}, where nn is a constant. What is the value of nn?
  1. Square x expression: To find the value of nn, we need to use the given solution form x=5+nx = 5 + \sqrt{n} and plug it into the quadratic equation x2−10x+14=0x^2 - 10x + 14 = 0.
  2. Simplify squared expression: First, let's square the expression for xx: (5+n)2=52+2×5×n+(n)2(5 + \sqrt{n})^2 = 5^2 + 2 \times 5 \times \sqrt{n} + (\sqrt{n})^2.
  3. Substitute xx into equation: Simplifying the squared expression gives us: 25+10n+n25 + 10\sqrt{n} + n.
  4. Combine like terms: Now, we substitute x=5+nx = 5 + \sqrt{n} into the quadratic equation: (5+n)2−10(5+n)+14=0(5 + \sqrt{n})^2 - 10(5 + \sqrt{n}) + 14 = 0.
  5. Further simplification: Substitute the simplified squared expression into the equation: 25+10n+n−10(5)−10n+14=025 + 10\sqrt{n} + n - 10(5) - 10\sqrt{n} + 14 = 0.
  6. Solve for n: Simplify the equation by combining like terms: 25+n−50+14=025 + n - 50 + 14 = 0.
  7. Solve for n: Simplify the equation by combining like terms: 25+n−50+14=025 + n - 50 + 14 = 0.Further simplification gives us: n−11=0n - 11 = 0.
  8. Solve for n: Simplify the equation by combining like terms: 25+n−50+14=025 + n - 50 + 14 = 0.Further simplification gives us: n−11=0n - 11 = 0.Solve for n: n=11n = 11.

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