= 2. Given the quadratic equation 3x2 – 8x + k = 0 has no real roots. Find the range of values of k.
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We want to find the values of a, b, c, and d such that the given matrix product is equal to a 2x2 identity matrix. We will solve a system of equations to find:
- b = 1
- a = -1
- c = -2
- d = -1
<h3>Presenting the equation:</h3>
Basically, we want to solve:
The matrix product will be:
Then we must have:
-b + 2 = 1
This means that:
b = 2 - 1 = 1
- b = 1
We also need to have:
a*b + 1 = 0
we know the value of b, so we just have:
a*1 + b = 0
- a = -1
Now the two remaining equations are:
-c + 2d = 0
a*c + d = 1
Replacing the value of a we get:
-c + 2d = 0
-c + d = 1
Isolating c in the first equation we get:
c = 2d
Replacing that in the other equation we get:
-(2d) + d = 1
-d = 1
- d = -1
Then:
c = 2d = 2*(-1) = -2
- c = -2
So the values are:
- b = 1
- a = -1
- c = -2
- d = -1
If you want to learn more about systems of equations, you can read: