The equation x^{2}-14x+31=63 is a quadratic equation. Quadratic equations are equations that contain at least one term that is squared, and are usually written in the form of ax^{2} + bx + c = 0, where a, b and c are constants, and x is an unknown quantity. In this equation, a = 1, b = -14, c = 31, and we need to determine the value of x.

One way to solve a quadratic equation is to use the quadratic formula. The quadratic formula is a formula that can be used to solve any quadratic equation. The formula states that the solution to a quadratic equation is given by x = (-b ± √b^{2}-4ac)/2a. In this equation, b^{2}-4ac = -444. Therefore, the solution to the equation x^{2}-14x+31=63 is given by x = (-(-14) ± √(-444))/2. This gives the solutions x = 11 and x = 7.

Another way to solve this equation is to use factoring. Factoring is a process of rewriting a mathematical expression in terms of its factors. In this equation, the expression can be written as (x-11)(x-7) = 0. This means that either x-11 = 0 or x-7 = 0. Therefore, the solutions to the equation x^{2}-14x+31=63 are x = 11 and x = 7.

In conclusion, the equation x^{2}-14x+31=63 can be solved using either the quadratic formula or the factoring method. Both methods give the same result, which is that the solutions to the equation are x = 11 and x = 7.

The equation x^{2}-14x+31=63 can be solved using either the quadratic formula or the factoring method. Both methods give the same result, which is that the solutions to the equation are x = 11 and x = 7.

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