Use the Fundamental Theorem of Algebra to determine which of the following equations have two solutions. Explain your reasoning in 5 sentences.
a. -9x + 4x^2 + 3 = 0
b. -2x^2 - 5 = 0
c. 5/7x^2 = 0
d. 0x^2 + 7x - 2 = 0
e. 0x^3 - x^2 + 2x + 4 = 0
The Fundamental Theorem of Algebra states that a polynomial equation of degree n has exactly n solutions, counting multiplicities.
a. The equation -9x + 4x^2 + 3 = 0 is a quadratic equation, which means it has at most two solutions. By the Fundamental Theorem of Algebra, it can have either zero, one, or two solutions. Therefore, further analysis is required to determine if it has two solutions.
b. The equation -2x^2 - 5 = 0 is also a quadratic equation. By the Fundamental Theorem of Algebra, it can have at most two solutions. Similar to (a), further analysis is needed to determine the number of solutions.
c. The equation 5/7x^2 = 0 is a quadratic equation with the coefficient of x^2 equal to zero, resulting in a linear equation. Linear equations have at most one solution, and by the Fundamental Theorem of Algebra, it can have at most one solution as well.
d. The equation 0x^2 + 7x - 2 = 0 is a linear equation since the coefficient of x^2 equals zero. Linear equations also have at most one solution. Therefore, it does not have two solutions.
e. The equation 0x^3 - x^2 + 2x + 4 = 0 is a cubic equation. According to the Fundamental Theorem of Algebra, it can have at most three solutions. However, in this case, we cannot conclude if it has exactly two solutions without further analysis.