Problem 55 - Elementary Algebra

Question

\(w^{2}-18 w+86=0\)

Step-by-Step Solution

Answer

The solutions are: \(w = 9 + i \sqrt{5}\) and \(w = 9 - i \sqrt{5}\).

Step 1: Identify the quadratic equation

The given equation is in the form of a standard quadratic equation: \[w^{2} - 18w + 86 = 0\].

Step 2: Determine coefficients

In the equation \(w^{2} - 18w + 86 = 0\), identify the coefficients: \(a = 1\), \(b = -18\), and \(c = 86\).

Step 3: Apply the quadratic formula

The quadratic formula is: \[w = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a}\]. Substitute the coefficients \(a = 1\), \(b = -18\), and \(c = 86\).

Step 4: Calculate the discriminant

The discriminant \(\Delta\) is calculated as:\[ \Delta = b^2 - 4ac \]. Substitute the values: \[ \Delta = (-18)^2 - 4 \times 1 \times 86 = 324 - 344 = -20 \].

Step 5: Determine the roots using the quadratic formula

Since the discriminant is negative \(\Delta = -20\), the roots are complex. Calculate the roots using the formula: \[w = \frac{18 \, \pm \, \sqrt{-20}}{2 \times 1} = \frac{18 \, \pm \, \, 2i \sqrt{5}}{2} \].

Step 6: Simplify the roots

The roots simplify to: \[w = 9 \, \pm \, i \sqrt{5}\].

Key Concepts

quadratic formula

The quadratic formula is used to find the solutions of quadratic equations of the form \[ax^2 + bx + c = 0\]. The formula itself is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].
By plugging in the coefficients from our specific equation, we can solve for the variables where the quadratic touches or crosses the x-axis.
Let’s break down the components:

• \(-b\): This flips the sign of the coefficient of the linear term.
• \(\sqrt{b^2 - 4ac}\): This is the discriminant, which helps us determine the nature of the roots.
• \(2a\): This normalizes everything, ensuring we divide the results properly by the coefficient found by the equation when it’s doubled.

If you're solving an equation like \(w^2 - 18w + 86 = 0\), substituting the coefficients \(a = 1\), \(b = -18\), and \(c = 86\) into the quadratic formula will yield the solutions.

complex roots

Complex roots occur when the solutions to a quadratic equation are not real numbers. This happens when the discriminant \(b^2 - 4ac\) is negative.
In these cases, the roots include imaginary numbers, which are typically expressed with an \(i\) symbol, representing \(\sqrt{-1}\).
Consider the quadratic equation \(w^2 - 18w + 86 = 0\). When we calculated the discriminant, it was \(-20\), indicating the roots are complex.
Using the quadratic formula, the roots are calculated as: \(w = \frac{18 \pm \sqrt{-20}}{2} = \frac{18 \pm 2i\sqrt{5}}{2}\). Simplifying further, the roots become \(9 \pm i\sqrt{5}\).
These complex roots show that the equation does not touch the real number line and instead exists in an entirely different number realm.

discriminant calculation

The discriminant of a quadratic equation determines the nature of its roots.
The discriminant (denoted as \(\Delta\)) is calculated using the formula: \(\Delta = b^2 - 4ac\).
Let's revisit the example \(w^2 - 18w + 86 = 0\). By substituting \(a = 1\), \(b = -18\), and \(c = 86\), we get: \(\Delta = (-18)^2 - 4 \times 1 \times 86 = 324 - 344\). This makes \(\Delta = -20\).
The sign of the discriminant determines the root type:

• If \(\Delta > 0\), there are two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root (a repeated root).
• If \(\Delta < 0\), there are two complex roots.

Since our discriminant is negative (\(\Delta = -20\)), we can confirm that the roots for \(w^2 - 18w + 86 = 0\) are indeed complex.

identifying coefficients

Before solving a quadratic equation using the quadratic formula, we need to identify its coefficients. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\).
In this example, the equation is \(w^2 - 18w + 86 = 0\).
Here’s how to identify coefficients:

• a: The coefficient of \(w^2\).In our equation, \(a = 1\).
• b: The coefficient of \(w\).For this equation, \(b = -18\).
• c: The constant term.Here, \(c = 86\).

Recognizing these values correctly is crucial because they are substituted into the quadratic formula to find the roots.Reviewing these coefficients ensures accurate calculations and understanding of the equation’s structure.