Math Problem Statement

Use the square root property to solve the equation: (x + 5)^2 = -50

Solution

To solve the equation (x+5)2=50(x + 5)^2 = -50 using the square root property, we first need to take the square root of both sides.

Starting with the given equation: (x+5)2=50(x + 5)^2 = -50 Step 1: Apply the square root property. Take the square root of both sides. x+5=±50x + 5 = \pm \sqrt{-50}

Step 2: Notice that 50\sqrt{-50} involves a negative number under the square root, so we can express this as a complex number. Recall that 1=i\sqrt{-1} = i, where ii is the imaginary unit. x+5=±50ix + 5 = \pm \sqrt{50}i x+5=±25×2i=±52ix + 5 = \pm \sqrt{25 \times 2}i = \pm 5\sqrt{2}i

Step 3: Isolate xx by subtracting 5 from both sides. x=5±52ix = -5 \pm 5\sqrt{2}i

Thus, the solution to the equation is: x=5+52iorx=552ix = -5 + 5\sqrt{2}i \quad \text{or} \quad x = -5 - 5\sqrt{2}i

Would you like a more detailed breakdown or further explanation of any step? Here are some follow-up questions that might help:

  1. What is the square root property and how is it used to solve equations like this one?
  2. Can you explain why we end up with imaginary numbers in this equation?
  3. What is the significance of the complex number ii?
  4. How do you simplify the square root of negative numbers in general?
  5. How would you check the solutions to this equation?

Tip: Whenever you encounter a negative number under a square root, you know the solution will involve imaginary numbers, so be prepared to deal with ii.

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Math Problem Analysis

Mathematical Concepts

Square Root Property
Complex Numbers
Imaginary Unit

Formulas

√(a*b) = √a * √b
√-1 = i

Theorems

Square Root Property
Imaginary Numbers

Suitable Grade Level

Grades 9-12