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Chapter 8: Problem 12

Find all complex-number solutions. $$ 4 y^{2}=12 $$

Short Answer

Expert verified
y = \pm \sqrt{3}

Step by step solution

01

Isolate the quadratic term

Start by dividing both sides of the equation by 4 to isolate the quadratic term. \[ 4y^2 = 12 \] \[ y^2 = \frac{12}{4} \] \[ y^2 = 3 \]
02

Apply the square root

Take the square root of both sides. Remember to include both the positive and negative square roots.\[ y = \pm \sqrt{3} \]

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Key Concepts

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

quadratic equation
In mathematics, a quadratic equation is any equation that can be rearranged into the form \[ ax^2 + bx + c = 0 \] where x represents an unknown variable, and a, b, and c are constants with a 鈮� 0.
The most notable feature of these equations is the presence of the variable squared (hence quadratically). This means that the graph of the equation is a parabola.
For instance, in the given exercise, our equation is \[4y^2 = 12. \]
To solve for y, you'll need to perform operations to isolate the variable term, transform the equation into its standard form, and then apply specific mathematical techniques.
square root
The square root of a number x is a value that, when multiplied by itself, gives x. It's represented by the symbol \[\sqrt{}\].
For example: \[ \sqrt{9} = 3 \] because \[ 3 \times 3 = 9. \]
  • In the given exercise, we reach a point where we have \[y^2 = 3. \]
  • To solve for y, we take the square root of both sides of the equation. Remember, whenever you take the square root of both sides in a quadratic equation, you must consider both the positive and negative roots.

    • In the step-by-step solution, this is shown as: \[ y = \pm \sqrt{3} \]
    This is because \[ (\frac{{\sqrt{3}})\times (\sqrt{3}))\] give 3, and \(-\sqrt{3} \times -\sqrt{3} \) also give 3. Always remember this when solving equations involving square roots.
isolating the variable
Isolating the variable means getting the variable on one side of the equation by itself. This requires the application of inverse operations. Let's go through the steps in our exercise to illustrate this concept.
  • Start with: \[ 4y^2 = 12 \]
  • To isolate \(y^2\), divide both sides by 4: \[ y^2 = \frac{12}{4} \] which simplifies to \[ y^2 = 3. \]

    • Now that the variable term \(y^2\) is isolated, you can move forward with finding the values of y.
    These operations of isolating the variable are crucial for simplifying equations. They serve as foundational steps for solving quadratic equations and many other types in algebra.

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