Solve each equation in Exercises 83–108 by the method of your choice. x² - 4x + 29 = 0

Verified step by step guidance

1. Rewrite the quadratic equation in standard form, which is already given as $x^2\; -\; 4x\; +\; 29\; =\; 0$.
2. Identify the coefficients of the quadratic equation: $a\; =\; 1$, $b\; =\; -4$, and $c\; =\; 29$.
3. Use the quadratic formula, $x\; =\; \backslash frac\{-b\; \backslash pm\; \backslash sqrt\{b^2\; -\; 4ac\}\}\{2a\}$, to solve for $x$. Substitute the values of $a$, $b$, and $c$ into the formula.
4. Simplify the discriminant, $b^2\; -\; 4ac$. Calculate $(-4)^2\; -\; 4(1)(29)$ to determine if the roots are real or complex.
5. Substitute the simplified discriminant and other values into the quadratic formula, then simplify the expression to find the two solutions for $x$. If the discriminant is negative, express the solutions in terms of imaginary numbers.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to these equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for solving them effectively.

Discriminant

The discriminant of a quadratic equation, given by the formula D = b^2 - 4ac, helps determine the nature of the roots. If D > 0, there are two distinct real roots; if D = 0, there is exactly one real root (a repeated root); and if D < 0, the roots are complex (non-real). Analyzing the discriminant is crucial for predicting the type of solutions before attempting to solve the equation.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit defined as the square root of -1. When solving quadratic equations with a negative discriminant, the solutions will involve complex numbers, which are essential for fully understanding the solutions of such equations.