1. Find the largest integer \(k\) such that the equation \(5x^2 - kx + 8 = 0\) has no real solutions.
2. What values of \(j\) does the equation \((2x+7)(x-5) = -43 + jx\) have one real solution?
+310 Both of these involve using the discriminant \(b^2-4ac\).
1. Since there are no real solutions, \(b^2-4ac < 0\).
\((-k)^2-4*8*5 < 0\)
\(k^2 < 160\)
The largest integer k is 12.
2. First expand: \(2x^2-3x-35=-43+jx\)
Then combine like terms: \(2x^2-(3+j)x+8=0\)
Since there is one real solution, \(b^2-4ac = 0\)
\((-(3+j))^2-4*2*8=0\)
\(j^2+6j-55=0\)
\((j+11)(j-5)=0\)
So \(j=-11, 5\)
