Problem 58 - 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. $$x^{2}-10 x+22=0$$

Step-by-Step Solution

Answer

The solutions to the equation \(x^{2}-10 x+22=0\) are \(x = 5 + \sqrt{3}\) and \(x = 5 - \sqrt{3}\). Verification will need to be performed using a graphing utility.

Step 1: Identify the coefficients of the quadratic equation

The standard form of a quadratic equation is \(ax^{2} + bx + c = 0\). Comparing the given equation \(x^{2} - 10x + 22 = 0\) with the standard form, it can be seen that \(a = 1\), \(b = -10\), and \(c = 22\).

Step 2: Apply the Quadratic Formula

Substitute the values of \(a\), \(b\), and \(c\) into the Quadratic Formula: \(x = \frac{-(-10) ± \sqrt{(-10)^{2} - 4*1*22}}{2*1} = \frac{10 ± \sqrt{100 - 88}}{2} = \frac{10 ± \sqrt{12}}{2} = \frac{10 ± 2\sqrt{3}}{2} = 5 ± \sqrt{3}\). So, the solutions to the equation are \(x = 5 + \sqrt{3}\) and \(x = 5 - \sqrt{3}\).

Step 3: Verify the solutions graphically

You need to use a graphing utility to graph the equation \(y = x^{2} - 10x + 22\). Ideally, the graph will intersect the x-axis at \(x = 5 + \sqrt{3}\) and \(x = 5 - \sqrt{3}\), verifying the solutions.

Key Concepts

Quadratic Formula

The Quadratic Formula is an essential tool for solving quadratic equations. A quadratic equation is typically in the form: \[ ax^2 + bx + c = 0 \] The formula allows you to find the solutions without factoring or graphing. It is expressed as: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.
• The \(\pm\) symbol means there are typically two solutions.
• The term \(b^2 - 4ac\) is called the discriminant.

In our exercise, we substituted \(a = 1\), \(b = -10\), and \(c = 22\) into the formula:\[x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \times 1 \times 22}}{2 \times 1}\]The results are \(x = 5 + \sqrt{3}\) and \(x = 5 - \sqrt{3}\). These are the solutions of the equation.

Coefficient Identification

Before using the Quadratic Formula, it's crucial to identify the coefficients correctly. In any quadratic equation, the coefficients \(a\), \(b\), and \(c\) correspond to specific terms:

• \(a\) is the coefficient of \(x^2\).
• \(b\) is the coefficient of \(x\).
• \(c\) is the constant term.

In our example eqution: \[x^2 - 10x + 22 = 0\]we find:

• \(a = 1\) because it's the coefficient of \(x^2\).
• \(b = -10\) comes from the \(x\) term.
• \(c = 22\) as the constant term.

Understanding which numbers correspond to \(a\), \(b\), and \(c\) is essential for correctly using the Quadratic Formula.

Graphical Solution Verification

Graphical verification provides a visual way to confirm the solutions you found algebraically. Once you have your solutions from the Quadratic Formula, you can use graphing software or a calculator:

• Plot the quadratic equation as \(y = x^2 - 10x + 22\).
• Check where the parabola crosses the x-axis.

The x-intercepts are the solutions to the equation. For our equation:

• The graph should intersect the x-axis at \(x = 5 + \sqrt{3}\) and \(x = 5 - \sqrt{3}\).
• If it does, your solutions are verified.

Seeing the parabola intersect at the calculated points confirms that your algebraic solutions are correct.