Problem 38 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}+8 x+13=0$$

Step-by-Step Solution

Answer

The solutions are \( x = -4 + \sqrt{3} \) and \( x = -4 - \sqrt{3} \).

Step 1: Identify the Quadratic Equation

We are given a quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 8 \), and \( c = 13 \). Our goal is to solve for \( x \).

Step 2: Calculate the Discriminant

The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( D = b^2 - 4ac \). Substitute the values: \( D = 8^2 - 4(1)(13) = 64 - 52 = 12 \). Since \( D > 0 \), the equation has two distinct real solutions.

Step 3: Use Quadratic Formula to Find Solutions

The quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \) is used to find the roots. Substituting \( b = 8 \), \( D = 12 \), and \( a = 1 \), we get: \( x = \frac{-8 \pm \sqrt{12}}{2} \). Simplifying \( \sqrt{12} \), we get \( 2\sqrt{3} \). Thus, \( x = \frac{-8 \pm 2\sqrt{3}}{2} = -4 \pm \sqrt{3} \).

Step 4: Graphical Interpretation

Graph the quadratic equation \( y = x^2 + 8x + 13 \) using a graphing tool. The graph is a parabola that opens upwards. The roots \( x = -4 + \sqrt{3} \) and \( x = -4 - \sqrt{3} \) are where the curve intersects the x-axis. These graphical points verify our solutions.

Key Concepts

Understanding the Discriminant

The discriminant is a critical component of quadratic equations that tells us about the nature of the roots without solving the equation completely. For any quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( D \) is calculated using the formula \( D = b^2 - 4ac \). Here are some essential details to know about the discriminant:

• If \( D > 0 \), the quadratic equation has two distinct real roots, meaning it will cross the x-axis at two points.
• If \( D = 0 \), there is exactly one real root, also known as a repeated or double root. The parabola will touch the x-axis but not cross it.
• If \( D < 0 \), the equation has no real roots, indicating that the parabola does not intersect the x-axis at any point.

For our exercise, the discriminant was found to be \( D = 12 \), which is positive. This confirms there are two distinct real roots. Calculating and interpreting the discriminant is a crucial step in understanding what type of solutions we can expect from the quadratic.

Solving with the Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation, especially when factoring is complex or the roots are not immediately apparent. The formula is given by:

\[ x = \frac{-b \pm \sqrt{D}}{2a} \]

where \( a \), \( b \), and \( D \) are coefficients and the discriminant derived from \( ax^2 + bx + c = 0 \). This method is nearly universal for solving quadratic equations.

In our exercise, using the values \( a = 1 \), \( b = 8 \), and \( D = 12 \), we substitute these into the formula:

• Calculate \( -b = -8 \)
• Compute \( \sqrt{12} = 2\sqrt{3} \)
• Substitute into the formula, simplifying to \( x = -4 \pm \sqrt{3} \)

These calculations provide the exact solutions for the quadratic equation. Utilizing the quadratic formula ensures precision, and it is applicable to any type of quadratic equation.

Graphical Solutions of Quadratic Equations

Graphical solutions provide a visual way to determine the roots of a quadratic equation. The graph of a quadratic equation is a parabola. Its shape, determined by the coefficient \( a \), can open upwards if \( a > 0 \) or downwards if \( a < 0 \). To find the roots graphically, we look for where the parabola intersects the x-axis.

• If the discriminant \( D > 0 \), the parabola will cross the x-axis at two distinct points, corresponding to the two real roots.
• If \( D = 0 \), the parabola touches the x-axis at exactly one point, indicating a repeated root.
• If \( D < 0 \), the parabola does not intersect with the x-axis, confirming there are no real roots.

In our specific example, the graph of \( y = x^2 + 8x + 13 \) illustrates the roots at \( x = -4 + \sqrt{3} \) and \( x = -4 - \sqrt{3} \), affirming our solutions. Graphs not only verify algebraic solutions but also help in understanding the geometric nature of equations.