Problem 63 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Quadratic Formula to solve the equation. Use a graphing utility to verify your solutions graphically. $$20 x^{2}-20 x+5=0$$

Step-by-Step Solution

Answer

The solution to the quadratic equation \( 20x^2 - 20x + 5 = 0 \) is \( x = 0.5 \), verified graphically by the x-intercept of the equation's graph.

Step 1: Identify a, b, and c from the equation

Given the quadratic equation in the form \( ax^2 + bx + c = 0 \), we can identify \( a = 20 \), \( b = -20 \), and \( c = 5 \). These values will be substituted into the quadratic formula.

Step 2: Substitute a, b, and c into the quadratic formula

Substitute \( a = 20 \), \( b = -20 \), and \( c = 5 \) into the quadratic formula. \( x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4*20*5}}{2*20} \).

Step 3: Simplify the expression under the square root

Next, calculate the value under the root in the quadratic formula, finding the value of the discriminant \( B^2 - 4AC \). Thus, the equation becomes \( x = \frac{20 \pm \sqrt{400 - 400}}{40} \).

Step 4: Calculate the roots of the equation

You'll find that the expression under the square root simplifies to zero, so both roots of the quadratic equation will be the same (the equation has a real repeated root). So \( x = \frac{20 \pm 0}{40} = \frac{20}{40} = 0.5 \). So, the solution to the equation is \( x = 0.5 \).

Step 5: Verify graphically

To visualize the solution, plot \( y = 20x^2 - 20x + 5 \) using a graphing utility. The x-intercepts of the graph (where y = 0) should correspond to the solution found, \( x = 0.5 \), confirming the validity of the solution.

Key Concepts

Discriminant

When solving quadratic equations using the quadratic formula, calculating the discriminant is a crucial step. The discriminant, represented as \( \Delta \), is part of the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). It is specifically the part under the square root: \( b^2 - 4ac \).
The value of the discriminant gives us key information about the nature of the roots of the equation. If \( \Delta > 0 \), the quadratic equation has two distinct real roots.

• When \( \Delta = 0 \), the equation has a repeated real root (also called a double root).
• If \( \Delta < 0 \), the equation has two complex roots (non-real roots).

In our example, the discriminant calculation is \( b^2 - 4ac = (-20)^2 - 4 \times 20 \times 5 = 400 - 400 = 0 \).
This tells us that there is one real repeated root.

Repeated Root

A repeated root in a quadratic equation occurs when the discriminant is zero. This results in the quadratic having only one unique solution, occurring twice. Mathematically, this means that the parabola represented by the equation just touches the x-axis at a single point rather than crossing it.
In our example, the equation \( 20x^2 - 20x + 5 = 0 \) ends up being simplified to have a repeated root.

• Both roots are \( x = \frac{20 \pm 0}{40} = 0.5 \).
• This translates to the vertex of the parabola being at \( x = 0.5 \).

Understanding the concept of repeated roots helps in visualizing the behavior of quadratic equations on a graph, as they exhibit a tangent with the x-axis.

Graphing Utility Verification

Using a graphing utility can be a powerful tool to verify solutions to quadratic equations. After solving the equation analytically, like using the quadratic formula, a graph helps confirm the solutions by visually displaying the behavior of the function.
To verify graphically, plot the function \( y = 20x^2 - 20x + 5 \). The graph of this quadratic will be a parabola.

• The curve should touch the x-axis at the repeated root \( x = 0.5 \).
• The point where it just touches (without crossing) is where the solution was verified to be a repeated root.

Graphing not only confirms the solution but also provides a clearer understanding of how the quadratic behaves across different values, reinforcing the concept of the repeated root.