Problem 64 - 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. $$9 x^{2}-18 x+9=0$$

Step-by-Step Solution

Answer

The solution to the quadratic equation \(9x^{2} - 18x + 9 = 0\) is \(x = 1\).

Step 1: Identify the coefficients

The coefficients of the given quadratic equation \(9x^{2} - 18x + 9 = 0\) are \(a = 9\), \(b = -18\), and \(c = 9\).

Step 2: Substitute the coefficients into the Quadratic Formula

Substitute the coefficients into the Quadratic Formula \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). This gives: \(x = \frac{-(-18) \pm \sqrt{(-18)^{2} - 4*9*9}}{2*9}\).

Step 3: Simplify to find the roots of the equation

Simplify the right hand side of the equation to find the roots: \(x = \frac{18 \pm \sqrt{324 - 324}}{18} = \frac{18 \pm 0}{18}\). The only solution to the quadratic equation is \(x = 1\).

Step 4: Verify the solutions graphically

To verify the solution graphically, plot the function \(f(x) = 9x^{2} - 18x + 9\). The graph should intersect the x-axis at \(x = 1\), thereby verifying the solution.

Key Concepts

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra, dealing with equations of the form \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). One of the most common methods to solve these equations is using the Quadratic Formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\).

To solve an equation using the Quadratic Formula, follow these steps:

• Identify the coefficients, \(a\), \(b\), and \(c\), from the equation.
• Substitute these values into the formula.
• Calculate the discriminant, \(b^{2} - 4ac\), to determine the number of solutions.
• Find the roots by simplifying the expression under the square root, and then adding and subtracting it to the negated value of \(b\), all divided by \(2a\).

The Quadratic Formula provides a guaranteed algebraic solution. For example, in the equation \(9x^{2} - 18x + 9 = 0\), the Quadratic Formula simplifies to \(x = 1\), indicating a single, repeated root.

Graphing Quadratic Functions

Graphing quadratic functions provides a visual representation of the solutions to a quadratic equation. The standard form of a quadratic function is \(f(x) = ax^{2} + bx + c\). The graph of this function is a parabola that opens upwards when \(a > 0\) and downwards when \(a < 0\).

Key features to identify on the graph are:

• The vertex, which is the highest or lowest point of the parabola, providing the maximum or minimum value of the function respectively.
• The axis of symmetry, a vertical line through the vertex dividing the parabola into symmetrical halves.
• The roots or x-intercepts, where the graph crosses the x-axis, representing the solutions to the equation \(f(x) = 0\).

To graph \(9x^{2} - 18x + 9\), we identify the vertex and see that since \(a > 0\), the parabola opens upwards. When graphed, the function touches the x-axis at the point \(x = 1\) signifying the root of the equation.

Roots of Quadratic Equations

The roots of quadratic equations, also known as the zeros or x-intercepts, are the values of \(x\) that satisfy the equation \(ax^{2} + bx + c = 0\). They are where the graph of the quadratic function intersects the x-axis.

To find the roots:

• First determine if they are real numbers by checking the discriminant \(b^{2} - 4ac\). If it's positive, the equation has two distinct real roots; if zero, it has exactly one real root (also known as a repeated root); and if negative, it has no real roots, implying the graph does not intersect the x-axis.
• Use the Quadratic Formula to calculate the exact values of the roots when they exist.

In our example, because the discriminant is zero (\(324 - 324 = 0\)), the equation \(9x^{2} - 18x + 9 = 0\) has a single, repeated root at \(x = 1\).

Using Graphing Utilities

Graphing utilities, such as graphing calculators or software applications, assist in visualizing quadratic functions and verifying solutions to quadratic equations. They are particularly useful when manual calculation is complex or when a real-time, accurate graph is required.

To use graphing utilities effectively:

• Enter the quadratic equation in its standard form into the utility.
• Adjust the viewing window to ensure the relevant features of the graph, like the vertex and roots, are visible.
• Analyze the graph to confirm the number and location of roots.

Using a graphing utility for our equation \(9x^{2} - 18x + 9 = 0\) shows a parabola with a vertex at the point \(x = 1\) and confirms the root found through the Quadratic Formula.