Solve each equation in Exercises 83–108 by the method of your choice. $x^{2} - 2 x = 1$

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Rewrite the given equation $x^2 - 2x = 1$ by moving all terms to one side to set the equation equal to zero: $x^2 - 2x - 1 = 0$.

Identify the coefficients in the quadratic equation $ax^2 + bx + c = 0$, where $a = 1$, $b = -2$, and $c = -1$.

Apply the quadratic formula to solve for $x$: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Substitute the values of $a$, $b$, and $c$ into the quadratic formula: $x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-1)}}{2(1)}$.

Simplify the expression under the square root (the discriminant) and then simplify the entire expression to find the two possible values of $x$.

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주요 개념

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Quadratic Equations

A quadratic equation is a polynomial equation of degree two, generally written as ax² + bx + c = 0. Solving such equations involves finding values of x that satisfy the equation, which can be done by factoring, completing the square, or using the quadratic formula.

Rearranging Equations

Before solving, it is important to rewrite the equation in standard form (ax² + bx + c = 0). This involves moving all terms to one side of the equation and simplifying, which sets the stage for applying solution methods effectively.

Methods of Solving Quadratic Equations

Common methods include factoring, completing the square, and the quadratic formula. Factoring works when the quadratic can be expressed as a product of binomials; completing the square transforms the equation into a perfect square trinomial; the quadratic formula provides a direct solution for any quadratic.

Solve each equation in Exercises 83–108 by the method of your choice. x2−2x=1x^2 - 2x = 1