Problem 57 - 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. $$x^{2}-9 x+19=0$$

Step-by-Step Solution

Answer

The solutions of the equation \(x^{2}-9x+19=0\) are \(x = (9 + sqrt(5)) / 2\) and \(x = (9 - sqrt(5)) / 2\).

Step 1: Identify a, b, and c

From the given equation \(x^{2}-9x+19=0\), it can be seen that a = 1, b = -9 and c = 19.

Step 2: Use the Quadratic Formula

The Quadratic Formula is used to find the solutions of the equation. Plugging a, b, and c into the formula gives \(x = [9 ± sqrt((-9)^2 - 4*1*19)] / (2*1)\), which simplifies to \(x = (9 ± sqrt(81 - 76)) / 2\). This simplifies further to \(x = (9 ± sqrt(5)) / 2\). So the solutions of the equation are \(x = (9 + sqrt(5)) / 2\) and \(x = (9 - sqrt(5)) / 2\).

Step 3: Verify Solutions Graphically

A graphing utility is used to graph the equation \(x^{2}-9x+19=0\). The values of x at which the graph intersects the x-axis are equivalent to the solution of the equation. The graph should intersect the x-axis at \(x = (9 + sqrt(5)) / 2\) and \(x = (9 - sqrt(5)) / 2\), thus visually verifying the solutions derived from the Quadratic Formula.

Key Concepts

Solving Quadratic Equations

Quadratic equations are expressions that take the form \( ax^2 + bx + c = 0 \). Solving them involves finding the values of \( x \) that make the equation true. One powerful method is using the Quadratic Formula, which applies to all quadratic equations. Here is the formula:

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

The components \( a \), \( b \), and \( c \) are coefficients from the equation, while \( \pm \) indicates there can be two solutions. Finding \( \sqrt{b^2 - 4ac} \) is essential, as it tells us if real solutions exist:

• The expression inside the square root, \( b^2 - 4ac \), is the discriminant.
• If it's positive, there are two real solutions.
• If it's zero, there's one real solution.
• If negative, the solutions are imaginary.

Using this formula, we calculate the exact solutions step by step, just like we did with \( x^2 - 9x + 19 = 0 \). This technique is universal and applicable for any quadratic equation.

Graphical Verification

Graphical verification allows us to check the solutions of a quadratic equation visually. By plotting the quadratic equation on a graph, we can see where the parabola intersects the x-axis. These intersection points correspond to the solutions found algebraically.
To verify graphically:

• Use a graphing tool or utility to plot the equation, such as a graphing calculator or software.
• Identify where the curve intersects the x-axis; these are the solutions.

In our example, \( x^2 - 9x + 19 = 0 \) graphically intersects at \( x = \frac{9 + \sqrt{5}}{2} \) and \( x = \frac{9 - \sqrt{5}}{2} \). This visual approach helps confirm our algebraic solution and understand it intuitively. Graphing provides a clear picture of the equation's behavior, helping to reinforce concepts and build a deeper understanding.

Algebraic Solutions

Algebraic solutions refer to solving equations using formulas and operations rather than approximations or tools. For quadratic equations, the Quadratic Formula is a primary method to find exact solutions. When applying it, you are essentially unraveling the equation to determine exact values for \( x \).
Key features of algebraic solutions include:

• Precision: They provide precise answers without the need for visual tools.
• Understanding: They require knowledge of symbols, operations, and formula manipulation.
• Reproducibility: The steps can be easily repeated and applied to other problems.

In our equation \( x^2 - 9x + 19 = 0 \), we carefully substituted values into the Quadratic Formula and simplified to find our two solutions. This method is essential for solving not just quadratics but can extend to polynomials and other complex equations, making it a cornerstone of algebra.