Use the Fundamental Theorem of Algebra to determine which of the following equations have two solutions. Explain your reasoning in 1-2 sentences.
a. -9x+4x^2+2=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
a. The equation -9x+4x^2+2=0 has two solutions because it is a quadratic equation, and the Fundamental Theorem of Algebra states that a quadratic equation can have at most two solutions.
b. The equation -2x^2-5=0 has two solutions because it is a quadratic equation, and the Fundamental Theorem of Algebra states that a quadratic equation can have at most two solutions.
c. The equation 5/7x^2=0 has only one solution because it factors into (5/7)x(x) = 0, so x=0 is the only solution.
d. The equation 0x^2+7x-2=0 has two solutions because it is a linear equation (degree of 1) and the Fundamental Theorem of Algebra states that a linear equation can have at most one solution.
e. The equation 0x^3-x^2+2x+4=0 has three solutions because it is a cubic equation (degree of 3) and the Fundamental Theorem of Algebra states that a cubic equation can have at most three solutions.
To determine which equations have two solutions, we can analyze the degree of each equation and apply the Fundamental Theorem of Algebra.
a. -9x+4x^2+2=0: This is a quadratic equation (degree 2), so it has two solutions according to the Fundamental Theorem of Algebra.
b. -2x^2-5=0: This is a quadratic equation (degree 2), so it has two solutions according to the Fundamental Theorem of Algebra.
c. 5/7x^2=0: This is a quadratic equation (degree 2), but when simplified, it becomes 5x^2=0. Since it has a single root at x=0, it does not have two solutions.
d. 0x^2+7x-2=0: This is a linear equation (degree 1), not a quadratic. It has one solution.
e. 0x^3-x^2+2x+4=0: This is a cubic equation (degree 3), and even though the leading coefficient is zero, it still has three solutions according to the Fundamental Theorem of Algebra.
To determine which of the equations have two solutions, we can use the Fundamental Theorem of Algebra, which states that a polynomial equation of degree n has exactly n solutions (counting multiplicities).
a. The equation -9x+4x^2+2=0 is a quadratic equation, so it can have at most two solutions.
b. The equation -2x^2-5=0 is a quadratic equation, so it can have at most two solutions.
c. The equation 5/7x^2=0 is a quadratic equation, so it can have at most two solutions.
d. The equation 0x^2+7x-2=0 is a linear equation, so it can have at most one solution (or none).
e. The equation 0x^3-x^2+2x+4=0 is a cubic equation, so it can have at most three solutions.
Therefore, equations a, b, and c can have two solutions, while equations d and e can have at most one and three solutions, respectively.