In the equation x^2-6x+c=0, find the values of c that will give two imaginary solutions.
a. c >= 5
b. c > 6
c. c < 6
d. c > 9
I figured out the equation with 6 (so I believe either b or c is the answer) being there, but is would it be > or <?
Bzzt. But thanks for playing...
two imaginary solutions means the discriminant is negative:
36-4c < 0
c > 9
Well, in the equation x^2-6x+c=0, the discriminant is the key to determining the nature of the solutions. If the discriminant is negative, then we have two imaginary solutions.
The discriminant of a quadratic equation ax^2+bx+c=0 is given by b^2-4ac.
So for the given equation x^2-6x+c=0, the discriminant is (-6)^2-4(1)(c), which simplifies to 36-4c.
To have two imaginary solutions, we need the discriminant to be negative.
If we solve the inequality 36-4c < 0, we get c > 9.
Therefore, the correct answer is d. c > 9.
To find the values of c that will give two imaginary solutions in the equation x^2-6x+c=0, we need to use the discriminant of the quadratic equation.
The discriminant (D) is given by the formula D = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation (ax^2 + bx + c = 0).
For two imaginary solutions, the discriminant must be negative. So, we need to find the values of c that make D negative.
In this case, the equation x^2-6x+c=0 has a = 1, b = -6, and c = c.
Substituting these values into the discriminant formula, we have D = (-6)^2 - 4(1)(c) = 36 - 4c.
For two imaginary solutions, we want D < 0. Therefore, we can set up the inequality: 36 - 4c < 0.
Simplifying the inequality, we have -4c < -36, which can be further simplified to c > 9.
Hence, the correct answer is option d. c > 9.
To find the values of c that will give two imaginary solutions in the equation x^2-6x+c=0, we can use the discriminant of the quadratic equation. The discriminant is given by the expression b^2-4ac, which determines the nature of the solutions.
In this case, the equation is x^2-6x+c=0, which can be rearranged into the standard quadratic form as ax^2+bx+c=0, where a = 1, b = -6, and c is the value we are looking for.
For the equation to have two imaginary solutions, the discriminant should be negative. Therefore, we need to calculate the discriminant using the values of a, b, and c:
Discriminant = (-6)^2 - 4(1)(c)
= 36 - 4c
Now, we need to compare this expression to zero (since we want a negative discriminant):
36 - 4c < 0
To simplify the inequality, subtract 36 from both sides:
-4c < -36
Next, divide both sides of the inequality by -4, remembering that dividing by a negative number flips the inequality sign:
c > 9
So, the correct answer is d. c > 9. This means that any value of c greater than 9 will result in two imaginary solutions for the equation x^2-6x+c=0.