Solve each equation by the method of your choice. √2 x2 + 3x - 2√2 = 0

Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

Problem 129

Chapter 2, Problem 129

Solve each equation by the method of your choice. √2 x² + 3x - 2√2 = 0

Verified step by step guidance

Rewrite the given equation:

\(\sqrt{2}\)x^2 + 3x - 2\(\sqrt{2}\) = 0

. This is a quadratic equation in standard form

ax^2 + bx + c = 0

, where

a = \(\sqrt{2}\)

b = 3

, and

c = -2\(\sqrt{2}\)

Use the quadratic formula to solve for

x

x = \(\frac{-b \pm \sqrt{b^2 - 4ac}\)}{2a}

. Substitute the values of

a

b

, and

c

into the formula.

Simplify the discriminant

b^2 - 4ac

: Compute

b^2

3^2 = 9

, and compute

4ac

4(\(\sqrt{2}\))(-2\(\sqrt{2}\))

. Simplify

4ac

to get

-16

. Then, calculate the discriminant:

b^2 - 4ac = 9 - (-16) = 25

Substitute the discriminant and other values into the quadratic formula:

x = \(\frac{-3 \pm \sqrt{25}\)}{2\(\sqrt{2}\)}

. Simplify

\(\sqrt{25}\)

5

, so the equation becomes

x = \(\frac{-3 \pm 5}{2\sqrt{2}\)}

Split the equation into two cases to find the two possible solutions for

x

: Case 1:

x = \(\frac{-3 + 5}{2\sqrt{2}\)}

, and Case 2:

x = \(\frac{-3 - 5}{2\sqrt{2}\)}

. Simplify each case to find the final solutions.

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Key Concepts

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

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to these equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for solving them effectively.

Square Roots

Square roots are the values that, when multiplied by themselves, yield the original number. In the context of the given equation, the presence of the square root indicates that we may need to isolate the variable or manipulate the equation to eliminate the square root. Mastery of square root properties is crucial for simplifying and solving equations involving them.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that can be multiplied together to obtain the original expression. In solving quadratic equations, factoring can provide a straightforward method to find the roots of the equation. Recognizing patterns and applying techniques such as the difference of squares or grouping are key skills in this process.

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Related Practice

Textbook Question

In Exercises 137–140, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The equation |x| = - 6 is equivalent to x = 6 or x = - 6.

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Textbook Question

When the sum of 1 and twice a negative number is subtracted from twice the square of the number, 0 results. Find the number.

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Textbook Question

When the sum of 6 and twice a positive number is subtracted from the square of the number, 0 results. Find the number.

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Textbook Question

Find b such that (7x + 4)/b + 13 = x has a solution set given by {- 6}.