Problem 59 - Precalculus Mathematics for Calculus

Question

Find the solutions of the inequality by drawing appropriate graphs. State each answer rounded to two decimals. $$x^{2} \leq 3 x+10$$

Step-by-Step Solution

Answer

The solution set for the inequality is \([-2, 5]\).

Step 1: Rearrange the Inequality

Start by rearranging the inequality to set it to zero. The inequality \(x^2 \leq 3x + 10\) becomes \(x^2 - 3x - 10 \leq 0\). This will help us understand where the parabola lies relative to the x-axis.

Step 2: Identify and Plot the Quadratic Function

Consider the function \(f(x) = x^2 - 3x - 10\). This function represents a parabola, and we need to find where this parabola is less than or equal to zero, i.e., below or on the x-axis.

Step 3: Calculate the Roots

To find where \(f(x) = 0\), we need to solve the quadratic equation using the quadratic formula: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. For \(a = 1\), \(b = -3\), \(c = -10\), we compute:\[b^2 - 4ac = (-3)^2 - 4(1)(-10) = 9 + 40 = 49\]\[x = \frac{3 \pm \sqrt{49}}{2} = \frac{3 \pm 7}{2}\]Thus, the roots are \(x = 5\) and \(x = -2\).

Step 4: Test Intervals between Roots

Since the parabola changes position at the roots, we must test intervals around them, such as \((-\infty, -2)\), \((-2, 5)\), and \((5, \infty)\), to see where the function is non-positive. Choose test points like \(x = -3\), \(x = 0\), and \(x = 6\). Substitute into the function:- For \(x = -3\), \(f(x) = (-3)^2 - 3(-3) - 10 = 9 + 9 - 10 = 8\), which is positive.- For \(x = 0\), \(f(x) = (0)^2 - 3(0) - 10 = -10\), which is negative.- For \(x = 6\), \(f(x) = (6)^2 - 3(6) - 10 = 36 - 18 - 10 = 8\), which is positive.

Step 5: Conclusion and Interval Notation

Since the function is negative between the roots, the inequality \(x^2 \leq 3x + 10\) holds for \(x \in [-2, 5]\). Hence, the solution set in interval notation is \([-2, 5]\).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally expressed in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The solutions to quadratic equations are the values of \( x \) which make the equation true. These solutions can be found through different methods: factoring, completing the square, using the quadratic formula, or graphing.

Key characteristics of quadratic equations include:

• The highest power of the variable is 2, which gives the equation its 'quadratic' namesake.
• The graph of a quadratic equation is a parabola, which can open upwards or downwards depending on the sign of \( a \).
• They can have zero, one, or two real solutions based on the discriminant \( b^2 - 4ac \).

For the inequality \( x^2 \leq 3x + 10 \), we've rearranged it to form \( x^2 - 3x - 10 \leq 0 \). This rearrangement shows us where the quadratic equation results in values less than or equal to zero, which ultimately indicate the solutions to our inequality.

Parabolas

The graph of a quadratic equation is called a parabola. It has a distinctive U-shaped curve and has several important attributes:

• Vertex: The turning point of the parabola. This is the highest or lowest point depending on whether the parabola opens downwards or upwards.
• Axis of Symmetry: A vertical line that runs through the vertex, splitting the parabola into two mirror-image halves.
• Direction: Determined by the sign of \( a \). If \( a > 0 \), the parabola opens upwards. If \( a < 0 \), it opens downwards.
• Roots: Points where the parabola intersects the x-axis, also known as x-intercepts or solutions of the equation.

For the equation \( f(x) = x^2 - 3x - 10 \), the graph is a parabola that opens upwards (since \( a = 1 > 0 \)). This graph helps us determine where the function is either negative or zero, identifying the interval of the solution to the inequality.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values, often solutions to inequalities. It uses brackets and parentheses to indicate which values are included.

• Square brackets \([ \; ]\) denote that an endpoint is included in the interval, often referred to as 'closed' intervals.
• Parentheses \(( \; )\) indicate that an endpoint is not included, known as 'open' intervals.
• A combination of these can describe a varied range of inclusivity for endpoints like \([a, b), [a, b], (a, b)\).

In this particular solution, because the parabola is below the x-axis between its roots at \(x = -2\) and \(x = 5\), we utilize interval notation to depict the solution set: \([-2, 5]\). This represents that all x-values within and including -2 to 5 satisfy the inequality \(x^2 \leq 3x + 10\).

Quadratic Formula

The quadratic formula is a reliable method to find the roots of any quadratic equation \( ax^2 + bx + c = 0 \). The roots can be calculated as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This powerful formula provides real and complex solutions, depending on the discriminant:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is one real root (a double root).
• If \( b^2 - 4ac < 0 \), the equation has two complex roots.

In the given inequality, to find the points where \( x^2 - 3x - 10 = 0 \), we set \( a = 1 \), \( b = -3 \), and \( c = -10 \). By substituting these into the formula, we obtained the roots: \( x = 5 \) and \( x = -2 \). These roots helped us identify the intervals for the parabola's behavior relative to the x-axis, crucial for solving the inequality.