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Exercises Polynomials and factoring

Example 9 from Precalculus, 3.2 Quadratic Functions

Solving a Quadratic Equation with the Quadratic Formula

Solve $x^{2} + x + 2 = 0.$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Let’s begin by writing the quadratic formula: $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} .$
When applying the quadratic formula, we identify the coefficients $a, b and c .$ For the equation $x^{2} + x + 2 = 0,$ we have $a = 1, b = 1, and c = 2.$ Substituting these values into the formula we have:

$\begin{array}{l} x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \\ = \frac{- 1 \pm \sqrt{1^{2} - 4 \cdot 1 \cdot (2)}}{2 \cdot 1} \\ = \frac{- 1 \pm \sqrt{1 - 8}}{2} \\ = \frac{- 1 \pm \sqrt{- 7}}{2} \\ = \frac{- 1 \pm i \sqrt{7}}{2} \end{array}$
The solutions to the equation are $\frac{- 1 + i \sqrt{7}}{2}$ and $\frac{- 1 - i \sqrt{7}}{2}$ or $\frac{- 1}{2} + \frac{i \sqrt{7}}{2}$ and $\frac{- 1}{2} - \frac{i \sqrt{7}}{2} .$

How did it go?

PreviousSubstituting an Imaginary Number in a Rational Function Next Identifying Power Functions

From Precalculus (1st edition) by OpenStax (Rice University), section 3.2, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.