Problem 15 - Algebra for College Students

Question

Solve each quadratic equation using the method that seems most appropriate to you. $$x^{2}-18 x+15=0$$

Step-by-Step Solution

Answer

The solutions are \( x = 9 + \sqrt{66} \) and \( x = 9 - \sqrt{66} \).

Step 1: Identify the Equation Type

The given equation \( x^2 - 18x + 15 = 0 \) is a quadratic equation because it is in the general form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -18 \), and \( c = 15 \).

Step 2: Decide on a Solving Method

For this quadratic equation, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This method is suitable because it can always find solutions for any quadratic equation.

Step 3: Determine the Discriminant

Calculate the discriminant using \( b^2 - 4ac \), where \( b = -18 \), \( a = 1 \), and \( c = 15 \). Thus, \( b^2 - 4ac = (-18)^2 - 4 \cdot 1 \cdot 15 = 324 - 60 = 264 \).

Step 4: Apply the Quadratic Formula

Substitute \( a = 1 \), \( b = -18 \), and \( c = 15 \) into the quadratic formula: \( x = \frac{-(-18) \pm \sqrt{264}}{2 \cdot 1} = \frac{18 \pm \sqrt{264}}{2} \).

Step 5: Simplify the Expression

First, simplify \( \sqrt{264} = \sqrt{4 \times 66} = 2\sqrt{66} \). Then continue simplifying the expression: \( x = \frac{18 \pm 2\sqrt{66}}{2} = 9 \pm \sqrt{66} \).

Step 6: Write the Solutions

The solutions to the equation \( x^2 - 18x + 15 = 0 \) are \( x = 9 + \sqrt{66} \) and \( x = 9 - \sqrt{66} \).

Key Concepts

Quadratic Formula

Quadratic equations can seem tricky, but the quadratic formula is a reliable method of finding solutions. It is represented by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula can solve any quadratic equation of the form \( ax^2 + bx + c = 0 \). Here’s how it works, step by step:

• Identify the values of \(a\), \(b\), and \(c\) from the equation. In our example, \(a = 1\), \(b = -18\), and \(c = 15\).
• Calculate the discriminant \(b^2 - 4ac\), which helps determine the nature of the roots.
• Substitute the values into the quadratic formula, solving for \(x\), using the results from the discriminant for further calculations.

Remember, the \(\pm\) symbol in the formula indicates you will generally find two solutions. This is because you can add or subtract the square root result from the value of \(-b\). After calculating, you'll find the roots of the equation which can be either real and distinct, real and equal, or complex.

Discriminant

The discriminant, noted as \(b^2 - 4ac\), is a crucial component in the quadratic formula. This value helps us determine the number and type of solutions for the quadratic equation.

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real double root.
• If the discriminant is negative, there are two complex roots.

In the problem \( x^2 - 18x + 15 = 0 \), the discriminant calculated is 264. Because it is positive, this means the equation has two distinct real roots.Knowing the discriminant value before using the quadratic formula can save you time, as it informs you what type of solutions to expect.

Simplifying Radicals

Radicals, or roots, often appear in solutions to quadratic equations, especially when using the quadratic formula. Simplifying radicals simplifies your final answer and makes it easier to interpret. Here's how you simplify, using \(\sqrt{264}\) as an example from our problem.

• Find the largest perfect square that divides the number under the radical. For 264, this is 4.
• Rewrite the radical as the product of the square root of the perfect square and another square root: \(\sqrt{264} = \sqrt{4 \times 66} = 2\sqrt{66}\).
• Simplify the expression under the main formula, such as \(x = \frac{18 \pm 2\sqrt{66}}{2}\).

Finally, you simplify further by dividing each part inside and outside of the radical by common factors, yielding the simplest form of the solution such as \(x = 9 \pm \sqrt{66}\). This not only makes the answer more presentable but also helps in understanding and using it in practical problems.