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Chapter 7: Problem 90

Find the imaginary solutions to each equation. $$3 x^{2}+4=0$$

Short Answer

Expert verified
The solutions are \( x = \frac{2i \text{sqrt}(3)}{3} \) and \( x = -\frac{2i \text{sqrt}(3)}{3} \).

Step by step solution

01

Isolate the quadratic term

Subtract 4 from both sides of the equation to isolate the quadratic term.\[ 3x^2 + 4 = 0 \]\[ 3x^2 = -4 \]
02

Divide by the coefficient of the quadratic term

Divide both sides by 3 to solve for \( x^2 \).\[ x^2 = \frac{-4}{3} \]
03

Solve for x

Take the square root of both sides. Remember to include both the positive and negative roots, and that the square root of a negative number involves the imaginary unit \( i \).\[ x = \frac{ i \times \text{sqrt}(4)}{ \text{sqrt}(3)} \]Simplify to get:\[ x = \frac{ 2i}{ \text{sqrt}(3)} \]}
04

Simplify the expression

Rationalize the denominator by multiplying the numerator and the denominator by \( \text{sqrt}(3) \).\[ x = \frac{2i \text{sqrt}(3)}{3} \]

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Key Concepts

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

isolate quadratic term
When solving quadratic equations, the first step is to isolate the quadratic term. This means you need to move all other terms to the opposite side of the equation. For example, if we start with the equation \(3x^2 + 4 = 0\), we subtract 4 from both sides. This step simplifies the equation to:\[3x^2 = -4\]. Isolating the quadratic term is crucial because it prepares the equation for the next steps, allowing us to solve for the variable more easily.
square root of negative number
After isolating the quadratic term and dividing by the coefficient, we often face taking the square root of a negative number. In the example equation \(3x^2 = -4\), we divide by 3:\[x^2 = \frac{-4}{3}\]. To solve for \(x\), we take the square root of both sides. The square root of a negative number involves the imaginary unit \(i\), which is defined as \(\text{sqrt}(-1) = i\). Therefore, the solution becomes:\[x = \frac{i \times \text{sqrt}(4)}{\text{sqrt}(3)}\]. Do not forget to include both the positive and negative roots, leading to:\[x = \frac{ \text{卤} 2i}{ \text{sqrt}(3)}\].
rationalize the denominator
The final step in solving some quadratic equations with imaginary solutions is to rationalize the denominator. This makes the solution cleaner and easier to understand. In the equation \(x = \frac{ 2i }{\text{sqrt}(3)}\), the denominator has a square root, which we want to eliminate. To do this, multiply both the numerator and the denominator by \(\text{sqrt}(3)\):\[x = \frac{2i \times \text{sqrt}(3)}{\text{sqrt}(3) \times \text{sqrt}(3)} = \frac{2i \text{sqrt}(3)}{3}\]. This process leaves us with a simpler, rationalized solution:\( x = \frac{2i \text{sqrt}(3)}{3} \).

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Most popular questions from this chapter

Work in a small group to write a formula that gives the side of a cube in terms of the volume of the cube and explain the formula to the other groups.

Solve each equation. $$(r-1)^{-2 / 3}=1$$

Simplify. $$\frac{\sqrt{3}}{\sqrt{6}-1}-\frac{\sqrt{3}}{\sqrt{6}+1}$$

Discussion Which one of the following expressions is not equivalent to the others? a) \((\sqrt[3]{x})^{4}\) b) \(\sqrt[4]{x^{3}}\) c) \(\sqrt[3]{x^{4}}\) d) \(x^{4 / 3}\) e) \(\left(x^{1 / 3}\right)^{4}\)

Solve each equation. $$w^{-2 / 3}=4$$
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