Exercises 82–84 will help you prepare for the material covered in the next section. Solve: x²+4x−1=0

Verified step by step guidance

1. Identify the type of equation given. Here, the equation $x^2+\; 4x\; -\; 1\; =\; 0$ is a quadratic equation because it is a polynomial of degree 2.
2. Recall the quadratic formula, which is used to solve any quadratic equation of the form $ax^2\; +\; bx\; +\; c\; =\; 0$. The formula is $x\; =\; \backslash (\backslash frac\{-b\; \backslash pm\; \backslash sqrt\{b^2\; -\; 4ac\}\backslash )\}\{2a\}$.
3. Determine the coefficients from the equation: $a\; =\; 1$, $b\; =\; 4$, and $c\; =\; -1$.
4. Substitute the values of $a$, $b$, and $c$ into the quadratic formula: $x\; =\; \backslash (\backslash frac\{-4\; \backslash pm\; \backslash sqrt\{4^2\; -\; 4(1)(-1)\}\backslash )\}\{2(1)\}$.
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 second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on the coefficients.

Quadratic Formula

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

Discriminant

The discriminant, given by b² - 4ac, determines the nature of the roots of a quadratic equation. If positive, there are two distinct real roots; if zero, one real root; and if negative, two complex conjugate roots.