Textbook Question
Fill in the blank(s) to correctly complete each sentence.
The graph of ƒ(x) = √-x is a reflection of the graph of y = √x across the ___-axis.
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Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem R.6.70
Solve each quadratic equation using the quadratic formula. See Example 7.
-3x² + 6x + 5 = 0
Verified step by step guidance1
Identify the coefficients from the quadratic equation \(-3x^2 + 6x + 5 = 0\). Here, \(a = -3\), \(b = 6\), and \(c = 5\).
Recall the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Calculate the discriminant \(\Delta = b^2 - 4ac\) by substituting the values of \(a\), \(b\), and \(c\):
\[\Delta = (6)^2 - 4 \times (-3) \times 5\]
Evaluate the square root of the discriminant \(\sqrt{\Delta}\) to determine the nature of the roots (real and distinct, real and equal, or complex).
Substitute \(-b\), \(\sqrt{\Delta}\), and \(2a\) into the quadratic formula to write the two possible solutions for \(x\):
\[x = \frac{-6 \pm \sqrt{\Delta}}{2 \times (-3)}\]
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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 the discriminant.
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Quadratic Formula
The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation ax² + bx + c = 0. It uses the coefficients a, b, and c to calculate the roots, including complex solutions when the discriminant is negative.
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Discriminant
The discriminant, given by Δ = b² - 4ac, determines the nature of the roots of a quadratic equation. If Δ > 0, there are two distinct real roots; if Δ = 0, one real root; and if Δ < 0, two complex conjugate roots.
Related Practice
Textbook Question
CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. The numerical coefficient in the term -7yz² is ___________.
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Textbook Question
Solve each inequality. Give the solution set using interval notation. See Examples 8 and 9. -2x - 2 ≤ 1 + x
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Textbook Question
Solve each quadratic equation using the quadratic formula. See Example 7. x² - 2x - 2 = 0
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Textbook Question
Solve each quadratic equation using the square root property. See Example 6. x² = 16
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Textbook Question
Solve each quadratic equation using the zero-factor property. See Example 5. 25x² + 30x + 9 = 0