Solve $x^2-8 x+15<0$.
Select the intervals that must be tested. Choose three correct answers.
The given inequality is \(x^2 - 8x + 15 < 0\). To find the intervals that partition the number line for testing, we first need to determine the roots of the corresponding quadratic equation \(x^2 - 8x + 15 = 0\). We can factor the quadratic expression as follows: \((x - 3)(x - 5) = 0\). This tells us that the roots of the quadratic equation are \(x = 3\) and \(x = 5\).
The roots \(x = 3\) and \(x = 5\) divide the real number line into three distinct intervals. These intervals are where the sign of the quadratic expression \(f(x) = x^2 - 8x + 15\) remains constant. The three intervals are:
By testing a value in each of these intervals, we can determine where the inequality \(x^2 - 8x + 15 < 0\) holds true.
Comparing our identified intervals to the provided choices:
These three intervals are the standard regions one must test to solve a quadratic inequality by the method of intervals. Testing these regions reveals that the inequality is satisfied only in the interval \(3 < x < 5\). Therefore, the three intervals that must be tested are identified as (e), (c), and (b).
In engineering or physics, finding the intervals where a quadratic function is positive or negative is essential for optimization. For instance, if you are modeling the trajectory of a projectile with a quadratic function \(h(t) = -t^2 + 8t - 15\) representing height at time \(t\), you need to find the time intervals where the height is positive (above ground). Solving the inequality \(-t^2 + 8t - 15 > 0\) helps determine the time range during which the object remains in the air, allowing engineers to predict impact times or design landing mechanisms effectively.