Select the values that are solutions to the inequality $x^2+3 x-4>0$. Choose two correct answers.
To identify which of the given values (0, -6, 5, -2) satisfy the inequality \(x^2 + 3x - 4 > 0\), we first analyze the quadratic function \(f(x) = x^2 + 3x - 4\). The values of \(x\) that make the function positive are the solutions we are looking for.
To find the roots of the quadratic equation \(x^2 + 3x - 4 = 0\), we factor the expression. We look for two numbers that multiply to \(-4\) and add up to \(3\). These numbers are \(4\) and \(-1\). Thus, the quadratic factors as:
\[(x+4)(x-1) = 0\]
Setting each factor to zero, we find the roots:
\[x+4 = 0 \implies x = -4\]
\[x-1 = 0 \implies x = 1\]
These roots divide the number line into three intervals: \((-\infty, -4)\), \((-4, 1)\), and \((1, \infty)\). Since the coefficient of \(x^2\) is positive (\(1 > 0\)), the parabola opens upward. Therefore, the function \(f(x) = x^2 + 3x - 4\) is positive in the intervals \((-\infty, -4)\) and \((1, \infty)\).
Now we check each of the given choices to see if they fall into the intervals \((-\infty, -4)\) or \((1, \infty)\):
Thus, the values that satisfy the inequality are \(-6\) and \(5\).
Quadratic inequalities are useful in physics for determining ranges of motion. For example, if a projectile's height is modeled by a quadratic function, finding when the height is greater than a certain value involves solving a quadratic inequality. This helps predict when an object will be within a specific safety zone or reach a target range.