1. In the quadratic formula, −b±b2−4ac√2a, what is b2−4ac called?
A. The Radical
B. The Discriminant
C. The Solution
D. The Radical
2. Suppose that you are solving a quadratic equation and realize that the discriminant is equal to 7. Which of the following statements best describes the solutions to this quadratic equation?
A. There must be two real solutions.
B. There must be exactly one real solution.
C. There will be no solutions to this quadratic equation.
D. There must be two non-real solutions.
3. Which of the following statements is true regarding the solutions to the quadratic equation 5x^2+6x+2=0
A. This quadratic equation has exactly one real solution.
B. This quadratic equation two complex solutions.
C. This quadratic equation has two real solutions.
D. This quadratic equation has no solutions.
4. Does every quadratic equation have at least one real solution? Explain.
A. No. When the discriminant is less than zero, there are no real solutions.
B. Yes. When the discriminant is zero, there are two unique solutions. When the discriminant is not zero, there is exactly one solution.
C. Yes. When the discriminant is zero, there is exactly one solution. When the discriminant is not zero, there are two unique solutions.
D. No. When the discriminant is greater than zero, there are no real solutions.
5.
Identify the number of zeros of the polynomial x-x^2-x^6+8x^4
A. 4
B. 8
C. 2
D. 6
1. B. The Discriminant
2. A. There must be two real solutions.
3. B. This quadratic equation has two complex solutions.
4. A. No. When the discriminant is less than zero, there are no real solutions.
5. C. 2
1. B. The Discriminant
2. A. There must be two real solutions.
3. D. This quadratic equation has no solutions.
4. A. No. When the discriminant is less than zero, there are no real solutions.
5. C. 2
1. The expression b^2 - 4ac is called the discriminant. To compute the discriminant, you plug in the values of a, b, and c from the quadratic equation into the formula and simplify.
2. When the discriminant is equal to a positive number, like 7, there must be two real solutions to the quadratic equation. This means the equation will intersect the x-axis at two distinct points.
3. To determine the solutions to the quadratic equation 5x^2 + 6x + 2 = 0, you can calculate the discriminant. In this case, the discriminant is less than zero, so the equation has no real solutions. Instead, it has two complex solutions, when the discriminant is negative, as it is here.
4. Not every quadratic equation has at least one real solution. When the discriminant (b^2 - 4ac) is negative, there are no real solutions to the equation. This is because the quadratic curve does not intersect the x-axis. However, if the discriminant is zero, there is exactly one real solution, while if it's greater than zero, there are two unique real solutions.
5. The number of zeros of a polynomial is determined by finding the number of distinct x-values where the polynomial equals zero. In this case, the polynomial is x - x^2 - x^6 + 8x^4. To find the zeros, you set the polynomial equal to zero and solve for x. However, it's not easy to determine the exact roots of this polynomial without further calculations. Therefore, we can't determine the number of zeros from the given information.