determine the values of k for which the quadratic equation x^2=kx+9=o
a) 2 equal real roots
b) 2 distinct real roots
can i get help solving this and getting it explained?
Your question deals with the discriminant b^2 - 4ac
if b^2 - 4ac > 0 , you have 2 distinct real roots
if b^2 - 4ac = 0 , you have 1 real roots, (actually 2 equal real roots, but they count as 1)
if b^2 - 4ac < 0 , you have 2 imaginary or complex roots that are the conjugate of each other
So, after you fix your x^2=kx+9=o, run the above test.
I suspect you mean
x^2 + k x+ 9 = 0
x = [ -k +/- sqrt(k^2 - 36) / (2)
for real roots
k^2 - 36 must be >0
now what happens if k = 6 ?
Of course! Let's start by solving the quadratic equation for the values of k that will result in either two equal real roots or two distinct real roots.
a) To find the values of k that will give us two equal real roots, we can use the discriminant. The discriminant is the expression under the square root sign in the quadratic formula and its value determines the nature of the roots. In this case, we want the discriminant to be equal to zero.
The discriminant can be found using the formula: D = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation in the form ax^2 + bx + c = 0.
For the equation x^2 = kx + 9, we have:
a = 1 (coefficient of x^2)
b = -k (coefficient of x)
c = 9
Substituting these values into the discriminant formula:
D = (-k)^2 - 4(1)(9)
D = k^2 - 36
For two equal real roots, the discriminant D should be equal to zero:
k^2 - 36 = 0
Now we'll solve this equation for k:
k^2 = 36
k = ±√36
k = ± 6
Therefore, for two equal real roots, the values of k are k = 6 and k = -6.
b) Let's move on to finding the values of k for which the quadratic equation has two distinct real roots. For two distinct real roots, the discriminant D should be greater than zero.
Using the same quadratic equation, we have:
D = k^2 - 36
For two distinct real roots, D > 0:
k^2 - 36 > 0
Now, let's solve this inequality:
k^2 > 36
k > √36 (taking the square root of both sides)
k > 6 or k < -6
Therefore, for two distinct real roots, the values of k are k > 6 or k < -6.
To summarize:
a) For two equal real roots, k = 6 or k = -6.
b) For two distinct real roots, k > 6 or k < -6.