Question

unit: radical expressions and quadratic equations
progress:
the movement of the progress bar may be uneven because questions can be worth more or less (including zero
use the quadratic formula to solve the equation $x^2 - 2x = -9$.
$\circ$ $x = -1 + 2i\sqrt{2}$ or $x = -1 - 2i\sqrt{2}$
$\circ$ $x = 1 + 2i\sqrt{2}$ or $x = 1 - 2i\sqrt{2}$
$\circ$ $x = 1 + \sqrt{10}$ or $x = 1 - \sqrt{10}$
$\circ$ $x = -1 + \sqrt{10}$ or $x = -1 - \sqrt{10}$

Explanation:

Step1: Rewrite the equation in standard form

The standard form of a quadratic equation is \(ax^{2}+bx + c = 0\). Given the equation \(x^{2}-2x=-9\), we add 9 to both sides to get \(x^{2}-2x + 9=0\). Here, \(a = 1\), \(b=-2\), and \(c = 9\).

Step2: Recall the Quadratic Formula

The Quadratic Formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).

Step3: Calculate the discriminant \(b^{2}-4ac\)

Substitute \(a = 1\), \(b=-2\), and \(c = 9\) into the discriminant formula: \((-2)^{2}-4\times1\times9=4 - 36=-32\).

Step4: Substitute into the Quadratic Formula

\(x=\frac{-(-2)\pm\sqrt{-32}}{2\times1}=\frac{2\pm\sqrt{32}i}{2}\) (since \(\sqrt{-32}=\sqrt{32}i = 4\sqrt{2}i\)).
Simplify the fraction: \(\frac{2\pm4\sqrt{2}i}{2}=1\pm2\sqrt{2}i\).

Answer:

\(x = 1 + 2i\sqrt{2}\) or \(x = 1 - 2i\sqrt{2}\) (the second option: \(x = 1 + 2i\sqrt{2}\) or \(x = 1 - 2i\sqrt{2}\))