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

Ch. 1 - Equations and Inequalities

Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경

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Blitzer 8th Edition

Ch. 1 - Equations and Inequalities

문제 129

2장, 문제 129

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

검증된 단계별 안내

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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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.