Carcass wrote:
If -1/2 and -2 are the two roots of the quadratic equation \(ax^2 + 5x + b = 0\), then what is the value of a and b?
A. a = -2 and b = -2
B. a = -2 and b = -1
C. a = -2 and b = 1
D. a = 2 and b = 1
E. a = 2 and b = 2
Key concept: If \(x = -1/2\) and \(x = -2\) are roots of the equation, then those values SATISFY the equation.Let's plug \(x = -0.5\) and \(x = -2 \) into the equation...
If \(x = -0.5\), we get: \(a(-0.5)^2 + 5(-0.5) + b = 0\)
Simplify: \(0.25a - 2.5 + b = 0\)
Take this prettier, let's multiply both sides by \(4\) to get: \(a - 10 + 4b = 0\)
Add \(10\) to both sides:
\(a + 4b = 10\)If \(x = -2\), we get: \(a(-2)^2 + 5(-2) + b = 0\)
Simplify: \(4a - 10 + b = 0\)
Add \(10\) to both sides:
\(4a + b = 10\)We now have the following system of equations:
\(a + 4b = 10\)\(4a + b = 10\)Solve to get: \(a = 2\) and \(b = 2\)
Answer: E