Problem 91 Find the imaginary solutions to ... [FREE SOLUTION]

Chapter 7: Problem 91

Find the imaginary solutions to each equation. $$3 x^{2}+6=0$$

Short Answer

Expert verified

The solutions are $x = \pm i\sqrt{2}$.

Step by step solution

- Isolate the quadratic term

Start by subtracting 6 from both sides of the equation to isolate the term involving the variable. \[3x^2 + 6 - 6 = 0 - 6\] This simplifies to: \[3x^2 = -6\]

- Solve for the variable squared

Next, divide both sides by 3 to solve for $x^2$. \[\frac{3x^2}{3} = \frac{-6}{3}\] This simplifies to: \[x^2 = -2\]

- Find the square root of both sides

Take the square root of both sides of the equation to solve for $x$. Remember that the square root of a negative number involves an imaginary unit, $i$. \[x = \pm\sqrt{-2}\] This further simplifies to: \[x = \pm i\sqrt{2}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equations

A quadratic equation is a type of polynomial equation of the second degree. It typically has the form: \[ ax^2 + bx + c = 0 \] where 'a', 'b', and 'c' are constants, and 'x' represents the variable. In our exercise, the quadratic equation given is \[ 3x^2 + 6 = 0 \] . To solve quadratic equations, various methods can be used such as factoring, completing the square, or using the quadratic formula. In some cases, like in our exercise, the solutions can be imaginary when the equation involves taking the square root of a negative number.

Imaginary Numbers

Imaginary numbers arise when we take the square root of a negative number. The imaginary unit is represented by 'i'. By definition: $ i = \sqrt{-1} $. This allows us to express the square root of any negative number in terms of 'i'. For example: $ \sqrt{-4} = 2i $. In our exercise, when solving the final step $ x = \pm \sqrt{-2} $, we convert \sqrt{-2} to an imaginary number as $ i \sqrt{2} $.It's important to understand that imaginary numbers are used in mathematics to extend our number system and are often used in science and engineering fields.

Square Root

The square root of a number 'a' is a value that, when multiplied by itself, gives the number 'a'. For example, the square root of 9 is 3 because $ 3 \cdot 3 = 9 $. When dealing with negative numbers, we introduce imaginary numbers, since taking the square root of a negative number isn't possible within the set of real numbers. In our exercise, to find the solutions to $ x^2 = -2 $, we take the square root of both sides: $ x = \pm \sqrt{-2} $. This gives us the imaginary solutions involving 'i', resulting in: x = \pm i\sqrt{2}.

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91影视

Find the imaginary solutions to each equation. $$3 x^{2}+6=0$$

Short Answer

Step by step solution

- Isolate the quadratic term

- Solve for the variable squared

- Find the square root of both sides

Key Concepts

Quadratic Equations

Imaginary Numbers

Square Root

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