Answer:
To find the solutions of the equations \(f(x) = (x+5)(x-2)\) and \(g(x) = (2x+7)(x-2)\) and verify that \((-2,12)\) and \((2,0)\) are indeed solutions, you can follow these steps:
1. For \(f(x) = (x+5)(x-2)\):
To find the solutions, set \(f(x)\) equal to zero and solve for \(x\):
\((x+5)(x-2) = 0\)
Now, use the zero-product property, which means that if the product of two factors equals zero, at least one of those factors must be zero:
\((x+5) = 0\) or \((x-2) = 0\)
Solve each equation separately:
\(x+5 = 0\) gives \(x = -5\)
\(x-2 = 0\) gives \(x = 2\)
So, the solutions for \(f(x) = (x+5)(x-2)\) are \(x = -5\) and \(x = 2\).
When you substitute \(x = -2\) into \(f(x)\), you get:
\(f(-2) = (-2+5)(-2-2) = 3*(-4) = -12\)
Therefore, the solution \((-2, 12)\) is confirmed.
2. For \(g(x) = (2x+7)(x-2)\):
Follow the same process. Set \(g(x)\) equal to zero and solve for \(x):
\((2x+7)(x-2) = 0\)
Apply the zero-product property:
\((2x+7) = 0\) or \((x-2) = 0\)
Solve each equation separately:
\(2x+7 = 0\) gives \(2x = -7\), and then \(x = -7/2\)
\(x-2 = 0\) gives \(x = 2\)
So, the solutions for \(g(x) = (2x+7)(x-2)\) are \(x = -7/2\) and \(x = 2\).
When you substitute \(x = 2\) into \(g(x)\), you get:
\(g(2) = (2*2+7)(2-2) = (4+7)(0) = 11*0 = 0\)
Therefore, the solution \((2, 0)\) is confirmed.
These are the solutions for both equations, and they match the given solutions \((-2, 12)\) and \((2, 0)\).