Math Problem Statement

Simplify by using the imaginary unit i: (-10 ± √-72) / 2

Solution

To simplify the expression 10±722\frac{-10 \pm \sqrt{-72}}{2} using the imaginary unit ii:

  1. Start by addressing the square root of the negative value, 72\sqrt{-72}. Rewrite it as: 72=72i\sqrt{-72} = \sqrt{72} \cdot i Since 72\sqrt{72} simplifies further, we have: 72=362=62\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2} Therefore: 72=62i\sqrt{-72} = 6\sqrt{2}i

  2. Substitute this into the expression: 10±722=10±62i2\frac{-10 \pm \sqrt{-72}}{2} = \frac{-10 \pm 6\sqrt{2}i}{2}

  3. Split the numerator for simplification: 10±62i2=102±62i2\frac{-10 \pm 6\sqrt{2}i}{2} = \frac{-10}{2} \pm \frac{6\sqrt{2}i}{2}

  4. Simplify each term: 102=5and62i2=32i\frac{-10}{2} = -5 \quad \text{and} \quad \frac{6\sqrt{2}i}{2} = 3\sqrt{2}i

  5. Combine these terms: 5±32i-5 \pm 3\sqrt{2}i

Final Answer:

The simplified form is: 5+32i,532i-5 + 3\sqrt{2}i, \, -5 - 3\sqrt{2}i

Would you like additional details or explanations?


Here are five related questions to deepen understanding:

  1. How is the imaginary unit ii defined, and how is it used in calculations?
  2. What are the properties of square roots involving negative numbers?
  3. How can you simplify radicals such as 72\sqrt{72} step by step?
  4. What are the typical forms of complex numbers, and how do they appear in equations like this?
  5. How does the ± symbol affect the solution set of quadratic-like expressions?

Tip: Always simplify radicals as much as possible before proceeding with operations involving imaginary numbers!

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Math Problem Analysis

Mathematical Concepts

Imaginary Numbers
Complex Numbers
Radicals
Algebraic Simplification

Formulas

√(-a) = √a * i
Complex form a + bi

Theorems

Properties of Imaginary Numbers
Radical Simplification

Suitable Grade Level

Grades 9-10