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Solve for x. Enter the solutions from least to greatest. {:[(x-7)^(2)-25=0],[" lesser "x=◻],[" greater "x=◻]:}

Solve for x x . Enter the solutions from least to greatest.\newline(x−7)2−25=0 lesser x=□ greater x=□ \begin{array}{l} (x-7)^{2}-25=0 \\ \text { lesser } x=\square \\ \text { greater } x=\square \end{array}

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Q. Solve for x x . Enter the solutions from least to greatest.\newline(x−7)2−25=0 lesser x=□ greater x=□ \begin{array}{l} (x-7)^{2}-25=0 \\ \text { lesser } x=\square \\ \text { greater } x=\square \end{array}
  1. Identify Equation: Identify the equation to solve.\newlineThe given equation is (x−7)2−25=0(x-7)^2 - 25 = 0.
  2. Rewrite in Difference of Squares: Rewrite the equation in the form of a difference of squares.\newlineThe equation (x−7)2−25=0(x-7)^2 - 25 = 0 can be seen as a difference of squares since 2525 is the square of 55.
  3. Factor the Difference of Squares: Factor the difference of squares.\newlineThe difference of squares can be factored as (a2−b2)=(a−b)(a+b)(a^2 - b^2) = (a - b)(a + b).\newlineSo, (x−7)2−25=(x−7−5)(x−7+5)(x-7)^2 - 25 = (x-7 - 5)(x-7 + 5).
  4. Simplify the Factors: Simplify the factors.\newline(x−7−5)(x−7+5)(x-7 - 5)(x-7 + 5) simplifies to (x−12)(x−2)(x - 12)(x - 2).
  5. Set Factors Equal and Solve: Set each factor equal to zero and solve for xx. First, set x−12=0x - 12 = 0, which gives x=12x = 12. Then, set x−2=0x - 2 = 0, which gives x=2x = 2.
  6. Identify Lesser and Greater Solutions: Identify the lesser and greater solutions.\newlineSince 22 is less than 1212, the lesser xx is 22 and the greater xx is 1212.

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