Ch. R - Algebra Review

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. R - Algebra Review

Problem R.6.55

Chapter 1, Problem R.6.55

Solve each quadratic equation using the square root property. See Example 6. x² - 27 = 0

Verified step by step guidance

Start with the given quadratic equation: \(x^{2} - 27 = 0\).

Isolate the squared term by adding 27 to both sides: \(x^{2} = 27\).

Apply the square root property, which states that if \(x^{2} = a\), then \(x = \pm \sqrt{a}\).

Take the square root of both sides: \(x = \pm \sqrt{27}\).

Simplify the square root if possible by factoring out perfect squares: \(\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}\), so \(x = \pm 3\sqrt{3}\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Square Root Property

The square root property states that if x² = k, then x = ±√k. This method is used to solve quadratic equations that can be written in the form x² = constant by isolating x² and then taking the square root of both sides, considering both positive and negative roots.

Isolating the Variable Term

Before applying the square root property, the quadratic equation must be rearranged so that the x² term is isolated on one side of the equation. This often involves adding or subtracting constants to both sides to simplify the equation into the form x² = number.

Handling Negative Values Under the Square Root

If the constant on the right side of the equation is negative, the solution involves imaginary numbers because the square root of a negative number is not real. In such cases, the solutions are expressed using the imaginary unit i, where i² = -1.

Recommended video:

2:20

Imaginary Roots with the Square Root Property

Related Practice

Textbook Question

Solve each quadratic equation using the zero-factor property. See Example 5.

x² + 2x - 8 = 0

Textbook Question

Solve each quadratic equation using the zero-factor property. See Example 5. x² - 100 = 0

views

Textbook Question

Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. See Example 4. 4(x + 7) = 2(x + 12) + 2(x + 1)

views

Textbook Question

Solve each quadratic equation using the zero-factor property. See Example 5.

9x² - 12x + 4 = 0

Textbook Question

Solve each quadratic equation using the zero-factor property. See Example 5. 4x² - 4x + 1 = 0

views

Textbook Question

Solve each inequality. Give the solution set using interval notation. See Example 10. 10 ≤ 2x + 4 ≤ 16

views