Use the Quadratic Formula to solve the quadratic below: $\newline$ $x^{2}-5x+9=0$

Full solution

Q. Use the Quadratic Formula to solve the quadratic below:

\newline

x^{2}-5x+9=0

Identify coefficients: Identify the coefficients $a$ , $b$ , and $c$ in the quadratic equation $x^2 - 5x + 9 = 0$ . By comparing $x^2 - 5x + 9$ with the standard quadratic form $ax^2 + bx + c$ , we find that $a = 1$ , $b = -5$ , and $c = 9$ .
Apply Quadratic Formula: Apply the Quadratic Formula to find the roots of the equation. $\newline$ The Quadratic Formula is given by $(-b \pm \sqrt{b^2 - 4ac}) / (2a)$ . We will substitute $a = 1$ , $b = -5$ , and $c = 9$ into the formula.
Substitute values: Substitute the values of $a$ , $b$ , and $c$ into the Quadratic Formula. $\newline$ $(-(-5) \pm \sqrt{(-5)^2 - 4\cdot1\cdot9}) / (2\cdot1)$ simplifies to $(5 \pm \sqrt{25 - 36}) / 2$ .
Simplify square root: Simplify the expression under the square root. $25 - 36 = -11$ , so the expression becomes $(5 \pm \sqrt{-11}) / 2$ .
Complex roots: Since the value under the square root is negative, we will have complex roots. We can express $\sqrt{-11}$ as $\sqrt{11} \times \sqrt{-1}$ , where $\sqrt{-1}$ is the imaginary unit $i$ . The expression becomes $(5 \pm \sqrt{11}i) / 2$ .
Divide by $2$ : Simplify the expression by dividing both terms by $2$ . $\newline$ $(\frac{5}{2}) \pm (\frac{\sqrt{11}}{2})i$ is the simplified form of the roots in $a+bi$ form.
Write in $a+bi$ form: Write the roots in the simplest $a+bi$ form. $\newline$ The roots are $\frac{5}{2} + \left(\frac{\sqrt{11}}{2}\right)i$ and $\frac{5}{2} - \left(\frac{\sqrt{11}}{2}\right)i$ .