Problem 8 - Algebra for College Students

Question

First use the discriminant to determine whether the equation has two nonreal complex solutions, one real solution with a multiplicity of two, or two real solutions. Then solve the equation. $$8 x^{2}+18 x-5=0$$

Step-by-Step Solution

Answer

The equation has two real solutions: \(x = \frac{1}{4}\) and \(x = -\frac{5}{2}\).

Step 1: Identify coefficients

First, recognize the equation is in the standard quadratic form: \(ax^2 + bx + c = 0\).Here, \(a = 8\), \(b = 18\), and \(c = -5\).

Step 2: Calculate the discriminant

The discriminant for a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula: \(\Delta = b^2 - 4ac\).Substitute the coefficients into this formula:\(\Delta = (18)^2 - 4\cdot8\cdot(-5)\).

Step 3: Simplify the discriminant

Calculate the values:\(18^2 = 324\)\(4 \cdot 8 \cdot (-5) = -160\)Thus, \(\Delta = 324 - (-160) = 324 + 160 = 484\).

Step 4: Interpret the discriminant

Since the discriminant \(\Delta = 484\) is positive, the quadratic equation has two distinct real solutions.

Step 5: Solve the quadratic equation using the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\).Substitute the known values: \(x = \frac{-18 \pm \sqrt{484}}{2 \times 8}\).

Step 6: Calculate the solutions

Calculate \(\sqrt{484} = 22\), then substitute:\(x_1 = \frac{-18 + 22}{16} = \frac{4}{16} = \frac{1}{4}\) \(x_2 = \frac{-18 - 22}{16} = \frac{-40}{16} = -\frac{5}{2}\).

Key Concepts

Understanding the Discriminant

The discriminant is a numerical value that helps us determine the type of solutions for a quadratic equation without actually solving it. For any quadratic equation in the form of \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is calculated using the formula \( \Delta = b^2 - 4ac \). This single number tells us whether the solutions are real or complex and how many solutions there are.

• If \( \Delta > 0 \), there are two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution, but it is counted twice (a double root).
• If \( \Delta < 0 \), the solutions are complex numbers which are not real.

In the given exercise, we calculated \( \Delta = 484 \), which is greater than zero. This indicates that the quadratic equation has two distinct real solutions.

Identifying Real Solutions

Real solutions refer to the values of \( x \) for which the equation holds true, and they are part of the real number line. When the discriminant is positive, as in our exercise, the quadratic equation will have two different points where it crosses the x-axis, meaning two real solutions.
Let's summarize:

• The real solutions occur when the parabola represented by the quadratic touches or intersects the x-axis.
• A positive discriminant confirms these intersection points above and indicates that they are real numbers.

For the equation \( 8x^2 + 18x - 5 = 0 \), the solutions \( \frac{1}{4} \) and \( -\frac{5}{2} \) came from solving the set equations where the parabola crosses the x-axis, ensuring two points of intersection with these distinct real values.

Applying the Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations. For the standard quadratic equation \( ax^2 + bx + c = 0 \), the solutions for \( x \) can be found using the formula:\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]The process involves evaluating the discriminant \( \Delta \) first, as seen in our exercise. Once we know \( \Delta \), we substitute it along with the coefficients \( a \) and \( b \) into the formula.
Here, thanks to the positive discriminant, the solutions were:

• \( x_1 = \frac{-18 + 22}{16} = \frac{1}{4} \)
• \( x_2 = \frac{-18 - 22}{16} = -\frac{5}{2} \)

The quadratic formula solved the equation by giving precise calculations for these x-values where the graph intersects the x-axis, showcasing two real outputs derived from applying arithmetic operations and square roots correctly. This formula is reliable and often the quickest method to find accurate solutions for any quadratic equation.