Problem 67 - College Algebra

Question

Write a quadratic inequality whose solution set is \([-3,5]\)

Step-by-Step Solution

Answer

The quadratic inequality whose solution set is \[-3,5] is \(x^2 - 2x - 15 \leq 0\)

Step 1: Form the Quadratic Equation

Using the given roots -3 and 5, form a quadratic equation. Any quadratic equation can be expressed as \(f(x) = (x - a)(x - b)\) where a and b are roots. Here, this means our equation is \(f(x) = (x+3)(x-5)\)

Step 2: Determine the Correct Inequality to Use

The solutions of the inequality fall in the interval \[-3,5\], and this is the interval within which our function should yield negative results as it lies below the x-axis. Hence, we have to use a 'less than or equal to' inequality, so our quadratic inequality is \(f(x)=(x+3)(x-5) \leq 0\)

Step 3: Simplify the Quadratic Inequality

Simplify the inequality to obtain the final result. This gives us \(x^2 - 2x - 15 \leq 0\) as the quadratic inequality

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation that can be expressed in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The graph of a quadratic function is a parabola that opens upwards if \( a > 0 \) and downwards if \( a < 0 \)

To solve quadratic equations, one can use different methods, such as factoring, completing the square, or applying the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Understanding how to form a quadratic equation from given roots is crucial for solving various mathematical problems, including quadratic inequalities.

In the specific case of the exercise, the roots given are -3 and 5. By recognizing that these roots correspond to the points where the quadratic function intersects the x-axis, one can derive the quadratic equation \( f(x) = (x + 3)(x - 5) \) by setting up factors based on the roots.

Inequality Solutions

Solving inequality solutions involves finding the set of values for the variable that make the inequality true. Unlike equalities, inequalities do not have precise solutions but rather ranges or intervals of solutions. Inequalities are represented using symbols such as '\(<\)', '\(>\)', '\(\leq\)', and '\(\geq\)' to denote less than, greater than, less than or equal to, and greater than or equal to, respectively.

For the quadratic inequality in the exercise, we're interested in the set of values for \( x \) that will satisfy \( f(x) = (x + 3)(x - 5) \leq 0 \). The solution set indicates the interval where the parabola lies below or on the x-axis. To find this interval, we can test points within the given range and determine where the inequality holds true. For a close-ended interval \( [a, b] \), we include the endpoints, meaning that solutions at \( x = a \) and \( x = b \) are valid for the inequality.

Roots of a Quadratic

The roots of a quadratic are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). They represent the x-coordinates where the graph of the quadratic function intersects the x-axis. Also known as zeros, solutions, or x-intercepts, these roots can be real or complex numbers and are fundamental in graphing and understanding the behavior of quadratic functions.

In the context of the given exercise, the quadratic equation has the roots -3 and 5. The factored form \( (x + 3)(x - 5) \) is derived directly from these roots, corresponding to the points \( (-3, 0) \) and \( (5, 0) \) on the graph. It is important to recognize that the sign between the two factors will affect where the parabola lies relative to the x-axis, which in turn influences the inequality statement.