Problem 31 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$-4+9 x-2 x^{2}=0$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{2}\) and \(x = 4\).

Step 1: Rearrange the Equation

The given equation is \(-4 + 9x - 2x^2 = 0\). This is a quadratic equation and it can be rewritten in the standard form \(ax^2 + bx + c = 0\). Rearrange the terms to get \(-2x^2 + 9x - 4 = 0\), where \(a = -2\), \(b = 9\), and \(c = -4\).

Step 2: Use the Quadratic Formula

To find the real solutions, use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).Substitute \(a = -2\), \(b = 9\), and \(c = -4\) into the formula:\[ x = \frac{-9 \pm \sqrt{9^2 - 4(-2)(-4)}}{2(-2)} \].

Step 3: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\):\[ b^2 - 4ac = 9^2 - 4 \times (-2) \times (-4) = 81 - 32 = 49 \]. The discriminant is 49, which is positive, indicating two distinct real solutions.

Step 4: Solve for x

Substitute the discriminant back into the quadratic formula:\[ x = \frac{-9 \pm \sqrt{49}}{-4} \]\[ x = \frac{-9 \pm 7}{-4} \].Calculate the two possible values for \(x\):\( x_1 = \frac{-9 + 7}{-4} = \frac{-2}{-4} = \frac{1}{2} \), and \( x_2 = \frac{-9 - 7}{-4} = \frac{-16}{-4} = 4 \).

Step 5: Verify the Solutions Graphically

Graph the function \(f(x) = -2x^2 + 9x - 4\). The solutions to the equation \(-2x^2 + 9x - 4 = 0\) are the x-values where the graph intersects the x-axis. By plotting, you'll see the graph intersects the x-axis at \(x = \frac{1}{2} \) and \(x = 4\), confirming the solutions.

Key Concepts

Quadratic Formula

When dealing with quadratic equations, one of the most powerful tools we use is the Quadratic Formula. If you have a quadratic equation in the form \( ax^2 + bx + c = 0 \), the Quadratic Formula allows you to find the value of \( x \) by using:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula is derived from the process of completing the square. It's a universal solution path for quadratic equations, which means you can use it anytime the equation fits the quadratic form. The symbol \( \pm \) indicates that there are usually two solutions, one for addition and one for subtraction. Remember to identify the coefficients \( a \), \( b \), and \( c \) from the standard form to correctly substitute into the formula.

Discriminant

The discriminant is a crucial part of the Quadratic Formula, and it gives important information about the nature of the roots of the equation. The discriminant is simply the part under the square root in the Quadratic Formula:

• \( b^2 - 4ac \)

Depending on its value, we understand the nature of the solutions:

• If the discriminant is positive (> 0), there are two distinct real solutions.
• If it's zero, there's exactly one real solution, often referred to as a repeated or double root.
• If it's negative, there are no real solutions; instead, the roots are complex or imaginary numbers.

In our example, the discriminant was calculated as 49, which is positive, indicating two distinct real solutions.

Graphical Verification

Graphical verification is a visual way to confirm the solutions of a quadratic equation. In essence, this involves plotting the quadratic equation as a parabola and observing where it intersects the x-axis. Each intersection point corresponds to a solution of the equation.

• The x-values of intersection are the same values obtained by solving the quadratic equation algebraically.
• You can sketch the graph using a graphing calculator or software, or even by hand.

In our exercise, the graph of \( f(x) = -2x^2 + 9x - 4 \) shows intersections at \( x = \frac{1}{2} \) and \( x = 4 \). This matches the solutions we calculated, thereby confirming our results are accurate.

Real Solutions

Real solutions refer to the values of \( x \) that satisfy the quadratic equation under real number operations. In other words, these solutions are points where the parabola defined by the quadratic equation actually crosses or touches the x-axis.

• If a quadratic equation has real solutions, it means the graph will intersect the x-axis at these points.
• For our example \(-2x^2 + 9x - 4 = 0 \), we found real solutions \( x = \frac{1}{2} \) and \( x = 4 \).

Real solutions can help interpret many real-world contexts, such as projectile motion or optimization problems, where only real, tangible results make sense. Understanding when and why real solutions occur ties closely into comprehension of the discriminant and graphic representations.