Question

The set of values of \(p\) for which both the roots of the quadratic equation, \(4 x^{2}-20 p x+\left(25 p^{2}+15 p-66\right)=0\) are less than 2 will be (1) \((4 / 5,2)\) (2) \((2, \infty)\) (3) \((-1,4 / 5)\) \((4)(-\infty,-1)\)

Short answer

(4) (-∞, -1)

Step-by-step solution

Write down the quadratic equation

The given quadratic equation is \[4 x^{2} - 20 p x + \big(25 p^{2} + 15 p - 66\big) = 0.\]

Apply the condition for roots less than 2

For both roots to be less than 2, we must ensure that the value of the quadratic expression at \(x = 2\) is non-positive (i.e., \(f(2) \leq 0\)) and that the axis of symmetry \(x = \frac{b}{2a}\) is to the right of 2.

Plug in x = 2 into the equation

Substitute \(x = 2\) in the quadratic equation.\[4 (2)^{2} - 20 p (2) + \big(25 p^{2} + 15 p - 66\big) \leq 0\]

Simplify

Simplify the equation to get: \[16 - 40 p + 25 p^{2} + 15 p - 66 \leq 0\]Combine like terms:\[25 p^{2} - 25 p - 50 \leq 0\]

Factorize the quadratic inequality

Factorize the inequality:\[25(p^{2} - p - 2) \leq 0\]Then further factorize inside the parenthesis:\[25(p - 2)(p + 1) \leq 0\]

Solve the inequality and find intervals

Solve the inequality to get the intervals:The critical points are \(p = 2\) and \(p = -1\).Use a sign chart to find the interval where the product is less than or equal to zero:\(( -\text{∞}, -1 ] \cup [ 2, ∞ )\)But since we need values for which both roots are less than 2, we only consider \(p \leq -1\).

Conclusion based on constraints

The set of values satisfying the condition is \[p \leq -1\]: So, the final interval is \[(-\text{∞}, -1)\].

Key concepts

roots of quadratic equations

Understanding the concept of the roots of a quadratic equation is crucial in solving quadratic inequalities. A quadratic equation is a polynomial equation of the form \(ax^2 + bx + c = 0\). The solutions, or roots, of the quadratic equation are the values of \(x\) that make the equation true. These roots can be found using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation. The expression under the square root, \(b^2 - 4ac\), is called the discriminant, and it determines the nature of the roots:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If the discriminant is negative, the equation has no real roots, but two complex roots.

By understanding the roots, we can solve not only quadratic equations but also inequalities involving quadratic expressions.

inequality solving

Solving quadratic inequalities requires a good grasp of both quadratic equations and the principles of inequalities. A quadratic inequality is an inequality that involves a quadratic expression, such as \(ax^2 + bx + c \leq 0\). To solve a quadratic inequality:

• First, solve the corresponding quadratic equation \(ax^2 + bx + c = 0\) to find the critical points (which are the roots).
• Use these critical points to divide the number line into intervals.
• Test a point from each interval in the original inequality to determine if it satisfies the inequality.
• The solution will be the union of the intervals that satisfy the inequality.

For example, consider the quadratic inequality \(25(p - 2)(p + 1) \leq 0\) from our exercise. Here, \(p\) has critical points \(2\) and \(-1\). Testing intervals around these points will show where the quadratic expression is non-positive. Solving such inequalities helps us define ranges for variables under specific conditions.

critical points

In the context of quadratic equations and inequalities, critical points are the values of the variable that make the quadratic polynomial equal to zero. These points are crucial for solving inequalities because they help divide the number line into intervals. Each distinct interval can then be tested to see if it satisfies the inequality. For our exercise, the critical points were found by solving \(25(p - 2)(p + 1) = 0\), resulting in \(p = 2\) and \(-1\). These points are essential for determining the intervals on which the inequality \(25(p - 2)(p + 1) \leq 0\) can be solved. By analyzing the critical points and the intervals, we can accurately determine the range of values that satisfy the given conditions. This approach is not only fundamental for quadratic inequalities but also for a variety of mathematical applications involving polynomials.