Solve for x. x^(2)-x=1

Rewrite Equation: Rewrite the given equation in the standard form for a quadratic equation. The given equation is x^2 - x = 1. To solve for x, we need to rewrite the equation in the standard form of a quadratic equation, which is ax^2 + bx + c = 0. We can do this by subtracting 1 from both sides of the equation. x^2 - x - 1 = 0Use Quadratic Formula: Use the quadratic formula to find the solutions for x. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients from the standard form of the quadratic equation ax^2 + bx + c = 0. In our case, a = 1, b = -1, and c = -1.Calculate Discriminant: Calculate the discriminant (b^2 - 4ac). The discriminant is the part of the quadratic formula under the square root, which determines the nature of the roots. For our equation, the discriminant is: (-1)^2 - 4(1)(-1) = 1 + 4 = 5Apply Discriminant: Apply the discriminant to the quadratic formula. Now that we have the discriminant, we can find the solutions for x using the quadratic formula: x = (-(-1) ± √5) / (2 * 1) x = (1 ± √5) / 2Simplify Solutions: Simplify the solutions for x. We have two solutions based on the ± sign in the quadratic formula: x = (1 + √5) / 2 and x = (1 - √5) / 2 These are the two solutions to the equation x^2 - x = 1.

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Solve for $x$ . $\newline$ $x^{2}-x=1$

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\newline

x^{2}-x=1

Rewrite Equation: Rewrite the given equation in the standard form for a quadratic equation. $\newline$ The given equation is $x^2 - x = 1$ . To solve for $x$ , we need to rewrite the equation in the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$ . We can do this by subtracting $1$ from both sides of the equation. $\newline$ $x^2 - x - 1 = 0$
Use Quadratic Formula: Use the quadratic formula to find the solutions for $x$ . The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a$ , $b$ , and $c$ are the coefficients from the standard form of the quadratic equation $ax^2 + bx + c = 0$ . In our case, $a = 1$ , $b = -1$ , and $c = -1$ .
Calculate Discriminant: Calculate the discriminant $b^2 - 4ac$ . The discriminant is the part of the quadratic formula under the square root, which determines the nature of the roots. For our equation, the discriminant is: $(-1)^2 - 4(1)(-1) = 1 + 4 = 5$
Apply Discriminant: Apply the discriminant to the quadratic formula. $\newline$ Now that we have the discriminant, we can find the solutions for $x$ using the quadratic formula: $\newline$ $x = \frac{-(-1) \pm \sqrt{5}}{2 \times 1}$ $\newline$ $x = \frac{1 \pm \sqrt{5}}{2}$
Simplify Solutions: Simplify the solutions for $x$ . We have two solutions based on the $\pm$ sign in the quadratic formula: $x = \frac{1 + \sqrt{5}}{2}$ and $x = \frac{1 - \sqrt{5}}{2}$ These are the two solutions to the equation $x^2 - x = 1$ .