Problem 94 - Algebra for College Students

Question

Find all real or imaginary solutions to each equation. Use the method of your choice. $$x^{2}-8 x+17=0$$

Step-by-Step Solution

Answer

The solutions are \(4 + i\) and \(4 - i\).

Step 1: Identify the coefficients

The given equation is a quadratic equation in the standard form: \[ ax^2 + bx + c = 0 \]For the equation \[ x^{2}-8x+17=0 \], the coefficients are:\[ a = 1 \]\[ b = -8 \]\[ c = 17 \]

Step 2: Calculate the discriminant

The discriminant of a quadratic equation is found using the formula: \[ \text{Discriminant} = b^2 - 4ac \]Substitute the values of the coefficients:\[ \text{Discriminant} = (-8)^2 - 4(1)(17) \]\[ \text{Discriminant} = 64 - 68 \]\[ \text{Discriminant} = -4 \]

Step 3: Interpret the discriminant

Since the discriminant is less than zero, the equation has two complex (non-real) solutions.

Step 4: Find the solutions using the quadratic formula

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]First, compute the real part:\[ x = \frac{8 \pm \sqrt{-4}}{2} \]\[ x = \frac{8 \pm 2i}{2} \]Now, simplify:\[ x = 4 \pm i \]

Key Concepts

Discriminant

The discriminant is a crucial part of understanding quadratic equations. It helps determine the nature of the solutions we might expect. In any quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is given by the formula:

\[ \text{Discriminant} = b^2 - 4ac \]

Using this formula, you simply plug in the values of your coefficients \(a\), \(b\), and \(c\). Here's how:

• Calculate \( b^2 \).
• Calculate \( 4ac \).
• Subtract \( 4ac \) from \( b^2 \).

In our example, the discriminant calculation was:

\[ (-8)^2 - 4(1)(17) = 64 - 68 = -4 \]

Since the result is negative, this means there are no real solutions. Instead, we get complex solutions. A positive discriminant would mean two real solutions, and a discriminant of zero would mean one real solution (a repeated root).

Complex Solutions

Complex solutions come into play when the discriminant is negative. This happens because you cannot take the square root of a negative number within the real number system. Instead, we rely on imaginary numbers, specifically the unit \( i \), where \( i^2 = -1 \).

From our example, the discriminant is \( -4 \). So, we rewrite \( \text{Discriminant} = -4 \) as
\[ \text{Discriminant} = 4i^2 \]

Plugging this into the quadratic formula gives:

\[ x = \frac{-b \, \text{±} \, \text{√Discriminant}}{2a} \]

Replacing Discriminant with \(-4\) and computing further, we get:
\[ x = \frac{8 \, \text{±} \, 2i}{2} \]

Then, simplifying this expression:
\[ x = 4 \text{±} i \]

Thus, the solutions are

• \( x = 4 + i \)
• \( x = 4 - i \)

Complex solutions always come in conjugate pairs when derived from quadratic equations.

The Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation. Given an equation of the form \( ax^2 + bx + c = 0 \), the quadratic formula lets you find the solutions by substituting \(a\), \(b\), and \(c\) directly:

\[ x = \frac{-b \text{±} \text{√(b^2 - 4ac)}}{2a} \]

Here's how you can use it step-by-step:

• Calculate the discriminant \( b^2 - 4ac \).
• Find the square root of the discriminant. If it's negative, you'll deal with imaginary numbers.
• Substitute \(-b\), the square root of the discriminant, and \(2a\) into the formula.
• Solve for \( x \) by performing the arithmetic operations.

Let's see it with our example:

\[ x = \frac{8 \text{±} \text{√(-4)}}{2} \]

Simplifying \( \text{√(-4)} \) as \( 2i \), we get:

\[ x = \frac{8 \text{±} 2i}{2} \]

And further simplifying it results in:

\[ x = 4 \text{±} i \]

Thus, the quadratic formula helps us find that the solutions to our example equation are \( 4 + i \) and \( 4 - i \). This method works universally for any quadratic equation!