Real Solutions for x in a Complex Equation

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  • Thread starter mathdad
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In summary, the equation x - sqrt {x^2 - x} = sqrt {x}, where x > 0, has only one real solution which is x = 1. This was found by dividing through by sqrt {x} and substituting x = 1 into the original equation.
  • #1
mathdad
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Find all real solutions of the equation.

x - sqrt {x^2 - x} = sqrt {x}, where x > 0.

Since x cannot be 0, I divided through by sqrt {x} to simplify the equation.

I did a lot of math on paper and ended up with x = 1 as the answer.

I decided to substitute x = 1 into the original equation.

1 - sqrt {1^2 - 1} = sqrt {1}

1 - sqrt {1 - 1} = 1

1 - sqrt {0} = 1

1 - 0 = 1

1 = 1

What do you say?
 
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  • #2
RTCNTC said:
Find all real solutions of the equation.

x - sqrt {x^2 - x} = sqrt {x}, where x > 0.

Since x cannot be 0, I divided through by sqrt {x} to simplify the equation.

I did a lot of math on paper and ended up with x = 1 as the answer.

I decided to substitute x = 1 into the original equation.

1 - sqrt {1^2 - 1} = sqrt {1}

1 - sqrt {1 - 1} = 1

1 - sqrt {0} = 1

1 - 0 = 1

1 = 1

What do you say?
Looks good.

-Dan
 
  • #3
Good to know that I am right.
 

FAQ: Real Solutions for x in a Complex Equation

What does it mean to "find all real solutions"?

Finding all real solutions involves determining all possible numerical values of a given equation that satisfy the criteria of being real numbers. This means that the solutions must be valid and exist on the number line.

How do you find all real solutions for a given equation?

To find all real solutions, you must first simplify the equation and then use various algebraic methods such as factoring, the quadratic formula, or completing the square to solve for the variable. Then, you must check your solutions to ensure that they are indeed real numbers.

Is it possible to have no real solutions for an equation?

Yes, it is possible for an equation to have no real solutions. This occurs when the solutions result in imaginary or complex numbers, which cannot be plotted on the number line. For example, the equation x^2 + 1 = 0 has no real solutions.

What is the difference between real and imaginary solutions?

Real solutions are numerical values that exist on the number line, while imaginary solutions involve the use of the imaginary unit, i, which represents the square root of -1. Imaginary solutions cannot be plotted on the number line and are typically written in the form a + bi, where a and b are real numbers and i is the imaginary unit.

Why is it important to find all real solutions for an equation?

Finding all real solutions is important because it allows us to fully understand the behavior and properties of the given equation. It also helps us to accurately graph the equation and determine its intersections with other equations or lines. Additionally, in real-world applications, finding all real solutions can help us solve problems and make informed decisions.

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