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Ch. 1 - Equations and Inequalities
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 1 - Equations and InequalitiesProblem 100a
Chapter 2, Problem 100a

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

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1
Rewrite the quadratic equation in standard form, which is already given as x^2 - 4x + 29 = 0.
Identify the coefficients of the quadratic equation: a = 1, b = -4, and c = 29.
Use the quadratic formula, x = \(\frac{-b \pm \sqrt{b^2 - 4ac}\)}{2a}, to solve for x. Substitute the values of a, b, and c into the formula.
Simplify the discriminant, b^2 - 4ac. Calculate (-4)^2 - 4(1)(29) to determine if the roots are real or complex.
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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
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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.
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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.
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Related Practice
Textbook Question

Use interval notation to represent all values of x satisfying the given conditions. y = |2x - 5| + 1 and y > 9

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Textbook Question

Perform the indicated operations and write the result in standard form. (i98 - i94)/i49

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Textbook Question

Solve each equation in Exercises 83–108 by the method of your choice. x2=4x−7x^2 = 4x - 7

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Textbook Question

In Exercises 101–106, solve each equation. ∣x−5∣=2\(\left\)|\(\sqrt{x}\)-5\(\right\)|=2

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Textbook Question

Exercises 100–102 will help you prepare for the material covered in the next section. Factor: x2−6x+9x^2 - 6x + 9

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Textbook Question

In Exercises 91–100, find all values of x satisfying the given conditions.y1=6(2xx−3)2,y2=5(2xx−3),andy1 exceeds y2 by 6.y_1 = 6 \(\left\)( \(\frac{2x}{x - 3}\) \(\right\))^2, \(\quad\) y_2 = 5 \(\left\)( \(\frac{2x}{x - 3}\) \(\right\)), \(\quad\) \(\text{and}\) \(\quad\) y_1 \(\text{ exceeds }\) y_2 \(\text{ by }\) 6.

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