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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 75
Chapter 2, Problem 75

In Exercises 75–82, compute the discriminant. Then determine the number and type of solutions for the given equation. x2−4x−5=0x^2 - 4x - 5 = 0

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Identify the coefficients in the quadratic equation \(x^2 - 4x - 5 = 0\). Here, \(a = 1\), \(b = -4\), and \(c = -5\).
Recall the formula for the discriminant: \(\Delta = b^2 - 4ac\).
Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \(\Delta = (-4)^2 - 4(1)(-5)\).
Simplify the expression to find the discriminant value (do not calculate the final number here).
Use the discriminant to determine the number and type of solutions: if \(\Delta > 0\), there are two distinct real solutions; if \(\Delta = 0\), there is one real solution; if \(\Delta < 0\), there are two complex solutions.

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Key Concepts

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

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. It represents a parabola when graphed and can have zero, one, or two real solutions depending on its coefficients.
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Discriminant

The discriminant is the part of the quadratic formula under the square root, given by b² - 4ac. It determines the nature and number of solutions of a quadratic equation: if positive, two distinct real solutions; if zero, one real repeated solution; if negative, two complex solutions.
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Types of Solutions of Quadratic Equations

The solutions of a quadratic equation can be real or complex. Real solutions occur when the discriminant is zero or positive, indicating where the parabola intersects the x-axis. Complex solutions arise when the discriminant is negative, meaning the parabola does not cross the x-axis.
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Related Practice
Textbook Question

In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 5x + 7 = 2x + 7

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Textbook Question

Exercises 73–75 will help you prepare for the material covered in the next section. Rationalize the denominator: (7 + 4√2)/(2 - 5√2).

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Textbook Question

In Exercises 61–76, solve each absolute value equation or indicate that the equation has no solution. |2x - 1| + 3 = 3

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Textbook Question

Solve each equation by the method of your choice. 3x2−7x+1=03x^2-7x+1 =0

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Textbook Question

Solve each equation by the method of your choice. (x−3)2−25=0(x-3)^2 - 25 = 0

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Textbook Question

In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 10x + 3 = 8x + 3

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