Sonia graphs the equations $y=-x^2+4 x$ and $y=x-4$ to solve the equation $-x^2+4 x=x-4$.
What are the solutions of $-x^2+4 x=x-4$?
The problem asks us to find the solutions to the equation \(-x^2 + 4x = x - 4\). To do this, we can set both expressions equal to a single side to form a quadratic equation of the form \(ax^2 + bx + c = 0\).
Starting with the given equation:
\[-x^2 + 4x = x - 4\]
Adding \(x^2\) and subtracting \(4x\) from both sides:
\[0 = x^2 - 4x + x - 4\]
\[x^2 - 3x - 4 = 0\]
This is a standard quadratic equation. We can solve it by factoring. We look for two numbers that multiply to \(-4\) and add up to \(-3\). Those two numbers are \(-4\) and \(1\).
Thus, we can write:
\[(x - 4)(x + 1) = 0\]
Setting each factor to zero gives us the potential solutions:
\[x - 4 = 0 \implies x = 4\]
\[x + 1 = 0 \implies x = -1\]
Therefore, the solutions to the equation are \(x = 4\) and \(x = -1\).
We verified the solutions by substituting them back into the original equation \(-x^2 + 4x = x - 4\).
For \(x = 4\):
Left side: \(-(4)^2 + 4(4) = -16 + 16 = 0\)
Right side: \(4 - 4 = 0\)
Since \(0 = 0\), \(x = 4\) is a solution.
For \(x = -1\):
Left side: \(-(-1)^2 + 4(-1) = -1 - 4 = -5\)
Right side: \(-1 - 4 = -5\)
Since \(-5 = -5\), \(x = -1\) is a solution.
The solutions are \(4\) and \(-1\).
Understanding how to find the intersection of a parabola and a line is very important in physics and economics. For example, if a company has a parabolic revenue function \(R(x) = -x^2 + 4x\) and a linear cost function \(C(x) = x - 4\), the values of \(x\) where the revenue equals the cost (the break-even points) are found by solving the equation \(-x^2 + 4x = x - 4\). Finding these points helps businesses determine the production level needed to become profitable.