Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$2 x^{2}-4 x=1$$

Step-by-Step Solution

Answer

The solutions are \( x = 1 + \frac{\sqrt{6}}{2} \) and \( x = 1 - \frac{\sqrt{6}}{2} \).

Step 1: Move all terms to one side

Start by moving all terms to one side of the equation to set the equation to zero: \[ 2x^2 - 4x - 1 = 0 \]

Step 2: Use the quadratic formula

Identify the coefficients from the quadratic equation \( ax^2+bx+c=0 \) where \( a=2 \), \( b=-4 \), and \( c=-1 \). Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 3: Calculate the discriminant

Calculate the discriminant \( b^2 - 4ac \) from the equation: \[ (-4)^2 - 4 \times 2 \times (-1) = 16 + 8 = 24 \] Since the discriminant is positive, there are two real solutions.

Step 4: Solve for x

Substitute the values into the quadratic formula to find \( x \): \[ x = \frac{-(-4) \pm \sqrt{24}}{2 \times 2} \] \[ x = \frac{4 \pm \sqrt{24}}{4} \]Simplify as:\[ x = \frac{4 \pm 2\sqrt{6}}{4} \]\[ x = 1 \pm \frac{\sqrt{6}}{2} \] So the solutions are \( x = 1 + \frac{\sqrt{6}}{2} \) and \( x = 1 - \frac{\sqrt{6}}{2} \).

Step 5: Support graphically

Graph the equation \( y = 2x^2 - 4x - 1 \) and observe where the graph crosses the x-axis. The solutions for \( x \) are the x-coordinates of these intersection points, which should match \( x = 1 + \frac{\sqrt{6}}{2} \) and \( x = 1 - \frac{\sqrt{6}}{2} \).

Key Concepts

Quadratic Formula

The quadratic formula is a critical tool for solving quadratic equations, which take the form \( ax^2 + bx + c = 0 \). Every quadratic equation can be solved using this formula, making it an incredibly versatile tool. The general quadratic formula is:

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

Here’s what each part means:

• \( a \), \( b \), and \( c \) are the coefficients of the terms from the equation.
• \( \pm \) means there will typically be two solutions—one for each sign.
• \( \sqrt{b^2 - 4ac} \) is the square root of the discriminant, which tells us about the nature of the roots.

To use the formula, you first need to identify \( a \), \( b \), and \( c \) from the equation. Substitute these values into the formula, and solve for \( x \). This formula gives you the solutions to the equation, provided they exist.

Discriminant

The discriminant is a key part of the quadratic formula that helps determine the nature of the solutions of a quadratic equation. It is represented by the formula \( \Delta = b^2 - 4ac \). Here's how it works:

• If \( \Delta > 0 \), the quadratic equation has two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution, meaning the parabola touches the x-axis at one point.
• If \( \Delta < 0 \), there are no real solutions. Instead, the solutions are complex numbers.

In the given problem, we calculated the discriminant as \( 24 \), which is greater than zero. This tells us that the quadratic equation has two distinct real solutions. This is consistent with what the quadratic formula will reveal when used properly.

Graphical Solutions

Graphical solutions of quadratic equations involve plotting the equation on a graph to visually find the solutions. The equation \( y = 2x^2 - 4x - 1 \) can be represented as a parabola on a coordinate plane.

• The x-values where the parabola crosses the x-axis are the solutions to the quadratic equation.
• If a parabola crosses the x-axis twice, the equation has two real solutions, which is what occurs in the given problem.

To sketch the graph, you can use a graphing calculator or software to plot the parabola. You'll observe the curve intersecting the x-axis at points corresponding to \( x = 1 + \frac{\sqrt{6}}{2} \) and \( x = 1 - \frac{\sqrt{6}}{2} \). Thus, these x-values are the solutions to the equation, verifying the solutions obtained using the quadratic formula.