Solve using the quadra $\newline$ a) $x^{2}=2 x+1$

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Algebra 2

Quadratic equation with complex roots

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Q. Solve using the quadra

\newline

x^{2}=2 x+1

Write Standard Quadratic Form: Write the equation in standard quadratic form. $\newline$ To find the roots of the equation, we need to write it in the form $ax^2 + bx + c = 0$ . Subtract $2x$ and $1$ from both sides to get the equation in standard form. $\newline$ $x^2 - 2x - 1 = 0$
Identify Coefficients: Identify the coefficients $a$ , $b$ , and $c$ . In the equation $x^2 - 2x - 1 = 0$ , the coefficients are: $a = 1$ , $b = -2$ , $c = -1$
Use Quadratic Formula: Use the quadratic formula to find the roots. $\newline$ The quadratic formula is given by $\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Substitute the values of $a$ , $b$ , and $c$ into the formula. $\newline$ $\frac{-(-2) \pm \sqrt{(-2)^2 - 4\cdot 1\cdot (-1)}}{2\cdot 1}$
Simplify Square Root: Simplify the expression under the square root. Calculate the discriminant $b^2 - 4ac$ . $(-2)^2 - 4\cdot 1\cdot (-1) = 4 + 4 = 8$
Substitute Discriminant: Substitute the discriminant back into the quadratic formula. $\frac{2 \pm \sqrt{8}}{2}$
Divide by $2$ : Simplify the square root of $8$ . $\sqrt{8}$ can be written as $\sqrt{4\times2}$ , which simplifies to $2\times\sqrt{2}$ . $(2 \pm 2\times\sqrt{2}) / 2$
Write Roots in Simplest Form: Simplify the expression by dividing by $2$ . $\newline$ Divide both terms in the numerator by $2$ . $\newline$ $1 \pm \sqrt{2}$
Write Roots in Simplest Form: Simplify the expression by dividing by $2$ . $\newline$ Divide both terms in the numerator by $2$ . $\newline$ $1 \pm \sqrt{2}$ Write the roots in simplest form. $\newline$ The roots of the equation are: $\newline$ $x = 1 + \sqrt{2}$ and $x = 1 - \sqrt{2}$