What are the solutions of $-x^2+4 x=x-4$ ?
We are tasked with solving the quadratic equation \(-x^2 + 4x = x - 4\). Our goal is to rearrange this equation into the standard quadratic form \(ax^2 + bx + c = 0\), find the roots of the equation, and then identify which of the given options matches these roots.
To bring the equation \(-x^2 + 4x = x - 4\) into standard form, we move all terms to one side of the equality:
\[-x^2 + 4x - x + 4 = 0\]
Combining like terms (\(4x - x = 3x\)), we obtain:
\[-x^2 + 3x + 4 = 0\]
To simplify, we multiply the entire equation by \(-1\):
\[x^2 - 3x - 4 = 0\]
This is a standard quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a=1\), \(b=-3\), and \(c=-4\).
We factor the quadratic expression \(x^2 - 3x - 4\). We need two numbers that multiply to \(-4\) and add up to \(-3\). These numbers are \(-4\) and \(1\). Therefore, we can write the equation as:
\[(x - 4)(x + 1) = 0\]
Setting each factor equal to zero, we find the solutions:
\(x - 4 = 0 \Rightarrow x = 4\)
\(x + 1 = 0 \Rightarrow x = -1\)
Thus, the solutions to the equation are \(x = 4\) and \(x = -1\). Comparing these with the given options, we identify that the correct choice is \(-1\) and \(4\).
Quadratic equations are widely used in physics and engineering to model motion. For instance, if you throw a ball in the air, its height \(h\) at time \(t\) can often be represented by a quadratic function like \(h(t) = -gt^2 + vt + h_0\). Solving for \(h(t) = 0\) helps you determine when the ball will hit the ground. By using the same factoring techniques demonstrated here, you can predict the trajectory of objects and understand the fundamental dynamics of movement in our everyday world.