Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Intro to Quadratic Equations

College Algebra

1. Equations & Inequalities

Intro to Quadratic Equations

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1:49 minutes

Problem 52

Textbook Question

Evaluate the discriminant for each equation. Then use it to determine the number and type of solutions. 16x² +3 = -26x

Verified step by step guidance

Rewrite the equation in standard quadratic form: \(16x^2 + 26x + 3 = 0\).

Identify the coefficients: \(a = 16\), \(b = 26\), \(c = 3\).

Use the discriminant formula: \(b^2 - 4ac\).

Substitute the coefficients into the formula: \(26^2 - 4 \times 16 \times 3\).

Evaluate the discriminant to determine the number and type of solutions: if positive, two real solutions; if zero, one real solution; if negative, 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 in the standard form of a quadratic equation 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.

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, typically expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. These equations can represent various real-world phenomena and are fundamental in algebra. Solving them often involves factoring, completing the square, or using the quadratic formula.

Standard Form

The standard form of a quadratic equation is a way of writing the equation as ax² + bx + c = 0. This format is essential for applying the quadratic formula and calculating the discriminant. To analyze the given equation, it must first be rearranged into this standard form, allowing for the identification of coefficients a, b, and c necessary for further calculations.