determine the value(s) of k which a quadratic equation x^2+kx+9=0 will have.
a) two equal real roots
b) 2 distinct roots
I tried to find the discriminant but couldn't get the right answer
why not show your work?
The discriminant is k^2-36
Now, what does that tell you about the number of roots?
To determine the values of k for which the quadratic equation x^2 + kx + 9 = 0 will have two equal real roots or two distinct roots, you need to consider the discriminant of the equation.
The discriminant is a mathematical expression that is found by substituting the coefficients of the quadratic equation into the formula b^2 - 4ac, where a, b, and c represent the coefficients of x^2, x, and the constant term, respectively.
For this particular equation, the coefficients are a = 1 (coefficient of x^2), b = k (coefficient of x), and c = 9 (constant term).
Now, substitute these values into the discriminant formula:
Discriminant = b^2 - 4ac
For two equal real roots, the discriminant should be equal to zero, which means:
0 = k^2 - 4(1)(9)
0 = k^2 - 36
For two distinct roots, the discriminant should be greater than zero, which means:
k^2 - 4(1)(9) > 0
k^2 - 36 > 0
To find the values of k, you need to solve these inequalities for k.
For two equal real roots:
k^2 - 36 = 0
Rearranging the equation,
k^2 = 36
Taking the square root of both sides,
k = ±6
Therefore, the values of k for which the equation will have two equal real roots are k = 6 and k = -6.
For two distinct roots:
k^2 - 36 > 0
Rearranging the inequality,
k^2 > 36
Taking the square root of both sides,
k > 6 or k < -6
Therefore, the values of k for which the equation will have two distinct roots are k > 6 or k < -6.
In summary:
a) For two equal real roots, k = 6 or k = -6.
b) For two distinct roots, k > 6 or k < -6.
Note: It's important to double-check the calculations to ensure accuracy.