Problem 55 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$4 x^{2}-20 x+25=0$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{5}{2}\), verified graphically by a tangent at this point.

Step 1: Recognize the Type of Equation

The given equation is a quadratic equation of the form \(ax^2 + bx + c = 0\). In this case, \(a = 4\), \(b = -20\), and \(c = 25\).

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \[D = (-20)^2 - 4(4)(25) = 400 - 400 = 0.\] Since the discriminant is zero, the equation has exactly one real solution.

Step 3: Solve for the Real Solution

When the discriminant is zero, the quadratic equation has a repeated real root. The solution is given by \(x = \frac{-b}{2a}\). Substitute \(b = -20\) and \(a = 4\) into the formula:\[x = \frac{-(-20)}{2 \times 4} = \frac{20}{8} = \frac{5}{2}.\]Thus, the quadratic equation has one real solution: \(x = \frac{5}{2}.\)

Step 4: Graphical Representation

To support the solution graphically, plot the quadratic function \(y = 4x^2 - 20x + 25\) on a coordinate plane. Since there is one real solution, the graph of the parabola will be tangent to the x-axis at \(x = \frac{5}{2}\). This tangency verifies the one real solution found algebraically.

Key Concepts

Discriminant

When dealing with quadratic equations of the form \(ax^2 + bx + c = 0\), the discriminant is a handy tool. It's calculated using the formula \(D = b^2 - 4ac\). The value of the discriminant tells us about the number and type of solutions the quadratic equation will have.

• If \(D > 0\), the equation has two distinct real solutions.
• If \(D = 0\), there's exactly one real solution, which is a repeated root.
• If \(D < 0\), the equation does not have real solutions, only complex ones.

This equation \(4x^2 - 20x + 25 = 0\) has a discriminant of zero (D = 0). This tells us that it not only has a real solution, but this solution is a double root, occurring at the same point on the number line.

Graphical Representation

Graphically, a quadratic equation is represented as a parabola on the coordinate plane. The shape and position of this parabola are determined by the coefficients of the quadratic equation.

When dealing with the function \(y = 4x^2 - 20x + 25\), it can be useful to visualize this on a graph to understand real solutions.

For our specific case:

• The parabola will either cross the x-axis, touch it at a single point, or miss it altogether.

Because the discriminant was zero (D = 0), the parabola touches the x-axis at exactly one point, corresponding to the single real solution. Using this representation, we observe:

• The parabola opens upwards since the coefficient of \(x^2\) is positive.
• There is a tangency at the x-axis, precisely at \(x = \frac{5}{2}\).

This graphical check serves as a visual confirmation that there is exactly one real solution, as determined by the algebraic method.

Real Solutions

A real solution for a quadratic equation is a value of \(x\) that satisfies the equation, meaning that when it is plugged into \(ax^2 + bx + c = 0\), the equation holds true. Real solutions are of great interest because they correspond to points where the parabola touches or crosses the x-axis.

In our equation, \(4x^2 - 20x + 25 = 0\), we found only one real solution: \(x = \frac{5}{2}\). Here's what happens:

• The parabola just grazes the x-axis at \(x = \frac{5}{2}\).
• This behavior is due to the fact that both sections of the parabola open away from the axis, not crossing it, just touching it at that single point.

This type of interaction specifies the only real solution found due to the quadratic's form and the properties of its coefficients. The real root represents the single point of contact with the x-axis, a fascinating geometric presentation of algebraic theory.