Problem 59 - Algebra and Trigonometry

Question

\(59-66\) . 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 solutions are \(x \in [-2, 5]\).

Step 1: Rearrange the Inequality

Let's start by rearranging the given inequality into a standard quadratic form. Move all terms to one side to get: \ \[ x^2 - 3x - 10 \leq 0 \] \

Step 2: Find the Roots of the Equation

We need to determine the points where the quadratic expression equals zero by solving \( x^2 - 3x - 10 = 0 \). Use the quadratic formula where \( a=1 \), \( b=-3 \), and \( c=-10 \): \ \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] \ Simplifying gives: \[ x = \frac{3 \pm \sqrt{(-3)^2 - 4 \times 1 \times (-10)}}{2 \times 1} \] \ \[ x = \frac{3 \pm \sqrt{9 + 40}}{2} \] \ \[ x = \frac{3 \pm \sqrt{49}}{2} \] \ \[ x = \frac{3 \pm 7}{2} \] \ So, the roots are \(x = 5\) and \(x = -2\).

Step 3: Analyze the Sign of the Quadratic

Now, we perform a sign analysis by testing intervals based on the roots \(-2\) and \(5\). We need to check: - \(-\infty, -2\) - \(-2, 5\) - \(5, \infty\)\Choose test points from each interval: - For \((-\infty, -2)\), let \(x = -3\), thus \((-3)^2 - 3(-3) - 10 = 9 + 9 - 10 = 8 > 0\); - For \((-2, 5)\), let \(x = 0\), thus \(0^2 - 3(0) - 10 = -10 < 0\); - For \((5, \infty)\), let \(x = 6\), thus \(6^2 - 3(6) - 10 = 36 - 18 - 10 = 8 > 0\).

Step 4: Determine Solution Interval

Since we are interested in when the inequality \(x^2 - 3x - 10 \leq 0\), and based on the previous analysis, the inequality holds true in the interval where the expression is less than or equal to zero: \ \(x \in [-2, 5]\). This includes the root values since the inequality is \(\leq\).

Step 5: Draw Graph to Visualize

Plot the quadratic function \(y = x^2 - 3x - 10\) on a graph. It is a parabola opening upwards. The graph intersects the x-axis at \(x = -2\) and \(x = 5\). Shade the region between these roots, on and below the x-axis, to represent the solution set of the inequality \(x^2 - 3x - 10 \leq 0\).

Key Concepts

Quadratic Equations

Quadratic equations are expressions in the form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown variable. The equation represents a parabola when graphed on a coordinate plane.
Quadratics are key in solving a variety of mathematical problems, such as finding the trajectory of an object or calculating profit maximizations.
Their standard form is crucial since it simplifies the process of solving and graphing, allowing a deeper understanding of its properties.

• **Standard Form:** Rearrange your inequality to replicate \( ax^2 + bx + c \) before solving.
• **Direction of Parabola:** A positive \( a \) results in an upward-opening parabola, while a negative \( a \) opens downward.

By understanding and identifying these forms, solving related problems becomes more manageable and intuitive.

Roots of Equations

Finding the roots of a quadratic equation means identifying the values of \( x \) where the equation equals zero. These roots are the x-intercepts of the graph of a quadratic function.

To compute these roots for the equation \( x^2 - 3x - 10 = 0 \), you can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, substituting \( a = 1 \), \( b = -3 \), and \( c = -10 \) we get,
\( x = 5 \) and \( x = -2 \).

• **Real Roots:** Determine when the graph crosses the x-axis. If \( b^2 - 4ac \geq 0 \), you have real roots.
• **Equal or Distinct Roots:** If \( b^2 - 4ac = 0 \), the roots are the same (tangent to the axis). If \( b^2 - 4ac > 0 \), the roots are distinct.

These roots are essential as they form the basis for the boundaries in interval testing.

Interval Testing

Interval testing is a vital technique in solving quadratic inequalities. Once you have the roots of \( x^2 - 3x - 10 \leq 0 \), you must test different intervals to find where the inequality holds.
The intervals, based on the roots \(-2\) and \(5\), are:

• **\((-\infty, -2)\):** Choose \( x = -3 \). When substituted, the result is greater than zero \((> 0)\).
• **\((-2, 5)\):** Choose \( x = 0 \). This results in a negative value \((< 0)\), showing this is the interval where the inequality is true.
• **\((5, \infty)\):** Choose \( x = 6 \). The output is again greater than zero \((> 0)\).

Interval testing helps identify where on the number line the original inequality \( x^2 - 3x - 10 \leq 0 \) is satisfied, leading to the answer \([-2, 5]\). These include root points since the inequality is \( \leq \).

Graphical Representation

Graphical representation is a powerful method to visualize quadratic inequalities. By plotting \( y = x^2 - 3x - 10 \), you can easily see the critical points and intervals of solution.

The graph forms a parabola that opens upwards, as the coefficient of \( x^2 \) is positive.

• **Intersections at the x-axis:** Occur at the roots \( x = -2 \) and \( x = 5 \).
• **Shaded Region:** Between these intersections, the graph is below or on the x-axis, representing solutions to the inequality.
• **Solution Interval:** Translate your testing into visual shading, helping confirm the interval \([-2, 5]\).

By shading or highlighting the relevant sections of the graph, students can better understand the range where the inequality \( x^2 - 3x - 10 \leq 0 \) holds. This is a clear, visual confirmation of the solution, offering an intuitive grasp of the concept.