A quadratic equation is shown.
$0=x^2-3 x-4$
Which value is a solution to this equation?
The given equation is a quadratic equation in the form \(x^2 - 3x - 4 = 0\). We are asked to identify which of the provided options—1, 3, 4, or 2—is a solution to this equation. A value is a solution if substituting it for \(x\) makes the equation true (i.e., results in 0).
We can find the solutions by factoring the quadratic expression \(x^2 - 3x - 4\). We are looking for two numbers that multiply to \(-4\) and add to \(-3\). These numbers are \(-4\) and \(1\). Thus, the equation can be factored as:
\[(x - 4)(x + 1) = 0\]
For this product to be zero, either \(x - 4 = 0\) or \(x + 1 = 0\). This gives two possible solutions: \(x = 4\) and \(x = -1\).
Comparing our calculated roots (\(x=4\) and \(x=-1\)) with the given options (1, 3, 4, 2), we see that 4 is a solution. We can verify this by substituting \(x = 4\) into the original equation:
\[4^2 - 3(4) - 4 = 16 - 12 - 4 = 16 - 16 = 0\]
Since the result is 0, the value 4 is indeed a solution. None of the other options (1, 3, or 2) satisfy the equation.
Quadratic equations are widely used in physics and engineering to model motion. For instance, if you throw a ball into the air, its height at a given time can often be represented by a quadratic equation. Finding the roots of such equations allows us to determine when the ball hits the ground (when the height is zero), which is essential for predicting the trajectory and landing point of objects.