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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 81a
Chapter 2, Problem 81a

Compute the discriminant. Then determine the number and type of solutions for the given equation. x2 - 3x - 7 = 0

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1
Identify the coefficients of the quadratic equation in the standard form ax^2 + bx + c = 0. For the equation x^2 - 3x - 7 = 0, the coefficients are: a = 1, b = -3, and c = -7.
Recall the formula for the discriminant, which is Δ = b^2 - 4ac. The discriminant helps determine the number and type of solutions for a quadratic equation.
Substitute the values of a, b, and c into the discriminant formula: Δ = (-3)^2 - 4(1)(-7).
Simplify the expression for the discriminant step by step: First, calculate (-3)^2, then compute 4(1)(-7), and finally subtract the results.
Interpret the discriminant value: If Δ > 0, the equation has two distinct real solutions. If Δ = 0, the equation has one real solution (a repeated root). If Δ < 0, the equation has two complex solutions.

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Key Concepts

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

Discriminant

The discriminant is a key component of the quadratic formula, given by the expression b² - 4ac for a quadratic equation in the form ax² + bx + c = 0. It helps determine the nature of the roots of the equation. If the discriminant is positive, there are two distinct real solutions; if it is zero, there is exactly one real solution; and if it is negative, there are two complex solutions.
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Quadratic Equation

A quadratic equation is a polynomial equation of degree two, typically expressed in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to a quadratic equation can be found using various methods, including factoring, completing the square, or applying the quadratic formula, which incorporates the discriminant.
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Types of Solutions

The types of solutions for a quadratic equation are classified based on the value of the discriminant. Real solutions occur when the discriminant is non-negative, leading to either two distinct real solutions or one repeated real solution. Complex solutions arise when the discriminant is negative, indicating that the roots are not real numbers and can be expressed in terms of imaginary numbers.
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Related Practice
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Solve each equation in Exercises 83–108 by the method of your choice. 2x2 - x = 1

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In Exercises 59–94, solve each absolute value inequality. - 2|x - 4| ≥ - 4

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Solve each absolute value inequality. - 4|1 - x| < - 16

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Solve each absolute value inequality. 5|2x + 1| - 3 ≥ 9

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Textbook Question

The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84. ∣x2−6∣=∣5x∣ |x^2 - 6| = |5x|

Textbook Question

The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars: If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84. ∣2x2−4∣=∣2x2∣|2x^2 - 4| = |2x^2|

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