Identify Equation Type: Identify the type of equation. The equation x^(2)=2x+1 is a quadratic equation in the form of ax^2 + bx + c = 0.Rewrite in Standard Form: Rewrite the equation in standard form by subtracting 2x and 1 from both sides to get x^2 - 2x - 1 = 0.Factor or Use Quadratic Formula: Attempt to factor the quadratic equation, if possible. In this case, the equation x^2 - 2x - 1 does not factor nicely, so we will use the quadratic formula instead. The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a), where a = 1, b = -2, and c = -1.Calculate Discriminant: Calculate the discriminant, which is the part under the square root in the quadratic formula: b^2 - 4ac. Here, it is (-2)^2 - 4(1)(-1) = 4 + 4 = 8.Insert Values into Formula: Insert the values of a, b, and c into the quadratic formula to find the solutions for x. x = (-(-2) ± √(8)) / (2*1) = (2 ± √(8)) / 2.Simplify Solutions: Simplify the solutions. Since √(8) = √(4*2) = 2√(2), the solutions are x = (2 ± 2√(2)) / 2.Final Simplification: Further simplify the solutions by dividing each term by 2. The final solutions are x = 1 ± √(2).

Calculus

Find derivatives using the chain rule I

Full solution

Q. Solve.

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x^{2}=2 x+1

Identify Equation Type: Identify the type of equation. The equation $x^{2}=2x+1$ is a quadratic equation in the form of $ax^2 + bx + c = 0$ .
Rewrite in Standard Form: Rewrite the equation in standard form by subtracting $2x$ and $1$ from both sides to get $x^2 - 2x - 1 = 0$ .
Factor or Use Quadratic Formula: Attempt to factor the quadratic equation, if possible. In this case, the equation $x^2 - 2x - 1$ does not factor nicely, so we will use the quadratic formula instead. The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 1$ , $b = -2$ , and $c = -1$ .
Calculate Discriminant: Calculate the discriminant, which is the part under the square root in the quadratic formula: $b^2 - 4ac$ . Here, it is $(-2)^2 - 4(1)(-1) = 4 + 4 = 8$ .
Insert Values into Formula: Insert the values of $a$ , $b$ , and $c$ into the quadratic formula to find the solutions for $x$ . $x = \frac{-(-2) \pm \sqrt{(8)}}{2 \cdot 1} = \frac{2 \pm \sqrt{(8)}}{2}$ .
Simplify Solutions: Simplify the solutions. Since $\sqrt{8} = \sqrt{4\cdot2} = 2\sqrt{2}$ , the solutions are $x = \frac{2 \pm 2\sqrt{2}}{2}$ .
Final Simplification: Further simplify the solutions by dividing each term by $2$ . The final solutions are $x = 1 \pm \sqrt{2}$ .