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Find one value of x that is a solution to the equation: {:[(x-2)^(2)-6(x-2)+5=0],[x=◻]:}

Find one value of x x that is a solution to the equation:\newline(x−2)2−6(x−2)+5=0x=□ \begin{array}{l} (x-2)^{2}-6(x-2)+5=0 \\ x=\square \end{array}

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Full solution

Q. Find one value of x x that is a solution to the equation:\newline(x−2)2−6(x−2)+5=0x=□ \begin{array}{l} (x-2)^{2}-6(x-2)+5=0 \\ x=\square \end{array}
  1. Expand and distribute: Simplify the equation by expanding (x−2)2(x-2)^2 and distributing −6-6 into (x−2)(x-2).
    (x−2)2(x-2)^2 can be expanded to x2−4x+4x^2 - 4x + 4.
    −6(x−2)-6(x-2) can be distributed to −6x+12-6x + 12.
    So the equation becomes x2−4x+4−6x+12+5=0x^2 - 4x + 4 - 6x + 12 + 5 = 0.
  2. Combine like terms: Combine like terms in the equation.\newlineCombine x2x^2, −4x-4x, and −6x-6x to get x2−10xx^2 - 10x.\newlineCombine 44, 1212, and 55 to get 2121.\newlineThe equation now is x2−10x+21=0x^2 - 10x + 21 = 0.
  3. Factor the quadratic equation: Factor the quadratic equation.\newlineWe need to find two numbers that multiply to 2121 and add up to −10-10.\newlineThe numbers −3-3 and −7-7 satisfy these conditions because (−3)×(−7)=21(-3) \times (-7) = 21 and (−3)+(−7)=−10(-3) + (-7) = -10.\newlineSo we can factor the equation as (x−3)(x−7)=0(x - 3)(x - 7) = 0.
  4. Solve for x: Solve for x by setting each factor equal to zero.\newlineFirst, set x−3=0x - 3 = 0, which gives us x=3x = 3.\newlineSecond, set x−7=0x - 7 = 0, which gives us x=7x = 7.\newlineSo the solutions to the equation are x=3x = 3 and x=7x = 7.

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