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a^(2)+1=0 How many distinct real solutions does the given equation have?

a2+1=0 a^{2}+1=0 \newlineHow many distinct real solutions does the given equation have?

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Full solution

Q. a2+1=0 a^{2}+1=0 \newlineHow many distinct real solutions does the given equation have?
  1. Analyze Equation: Analyze the equation a2+1=0a^{2}+1=0. The equation is a quadratic equation in standard form. We can see that it is not factorable using real numbers, so we will consider the properties of real numbers to determine the number of real solutions.
  2. Move Constant Term: Move the constant term to the other side of the equation.\newlineSubtract 11 from both sides to isolate the a2a^2 term.\newlinea2+1−1=0−1a^2 + 1 - 1 = 0 - 1\newlinea2=−1a^2 = -1
  3. Look for Solutions: Look for real solutions.\newlineThe left side of the equation a2a^2 represents the square of a real number. Since the square of any real number is non-negative, a2a^2 cannot be negative. Therefore, there are no real numbers aa such that a2=−1a^2 = -1.
  4. Conclude Solutions: Conclude the number of real solutions.\newlineSince there are no real numbers that satisfy the equation a2=−1a^2 = -1, the original equation a2+1=0a^2 + 1 = 0 has no real solutions.

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