Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x²+4x+6=0

Verified step by step guidance

1. Identify the type of equation given. Here, the equation is a quadratic equation of the form $x{2}^{}+\; 4x\; +\; 6\; =\; 0$, where the highest power of $x$ is 2.
2. Recall the quadratic formula, which is used to solve any quadratic equation $\mathrm{ax}{2}^{}+\mathrm{bx}+c=\; 0$. The formula is .
3. Identify the coefficients from the equation: $a\; =\; 1$, $b\; =\; 4$, and $c\; =\; 6$.
4. Substitute the values of $a$, $b$, and $c$ into the quadratic formula: .
5. Simplify the expression under the square root (the discriminant) and then simplify the entire expression to find the two possible values of $x$.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, generally written as ax² + bx + c = 0. It represents a parabola when graphed, and solving it means finding the values of x that satisfy the equation.

Completing the Square

Completing the square is a method used to solve quadratic equations by rewriting the equation in the form (x + p)² = q. This technique transforms the quadratic into a perfect square trinomial, making it easier to solve for x.

Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a direct way to find the roots of any quadratic equation ax² + bx + c = 0. It uses the coefficients to calculate the solutions, including complex roots when the discriminant is negative.