Problem 55 - Elementary Algebra

Question

\(b^{2}-6 b-16=0\)

Step-by-Step Solution

Answer

The solutions are b = 8 and b = -2.

Step 1: Identify the type of equation

The equation given is a quadratic equation in the form of ax^2 + bx + c = 0. Here, a = 1, b = -6, and c = -16.

Step 2: Apply the Quadratic Formula

The Quadratic Formula is given by \(b = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substituting a = 1, b = -6, and c = -16 into the formula.

Step 3: Calculate the discriminant

The discriminant is given by \(D = b^2 - 4ac\). For this equation, D = (-6)^2 - 4(1)(-16) = 36 + 64 = 100.

Step 4: Solve for the roots

Substitute the discriminant back into the Quadratic Formula: \(b = \frac{6 \pm \sqrt{100}}{2}\). Therefore, \(b = \frac{6 \pm 10}{2}\). This gives us two possible solutions: \(b = \frac{6 + 10}{2} = 8\) and \(b = \frac{6 - 10}{2} = -2\).

Key Concepts

Quadratic Formula

To solve a quadratic equation, we often use the Quadratic Formula. The standard form of a quadratic equation is given by \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants. The Quadratic Formula is:

\[ x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \]
This formula helps us find the roots of the quadratic equation. The term under the square root, \(b^2 - 4ac\), is called the discriminant. We'll discuss its importance in the next section. To apply the formula, plug in the values of \(a\), \(b\), and \(c\) from your quadratic equation and solve for \(x\).

For example, let's consider the equation from our exercise:

\[ b^2 - 6b - 16 = 0 \]
Here, \(a = 1\), \(b = -6\), and \(c = -16\). Substituting these values into the formula gives:
\[ x = \frac{6 \, \pm \, \sqrt{36 + 64}}{2} = \frac{6 \, \pm \, \sqrt{100}}{2} \]
We now have to solve the expression \(\frac{6 \, \pm \, 10}{2}\), which gives us two roots, 8 and -2.

Discriminant

The discriminant is a key part of the Quadratic Formula. It is represented by the term \(b^2 - 4ac\). The discriminant gives us information about the nature of the roots of the quadratic equation.

Depending on the value of the discriminant, we can determine the following:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root (or a repeated root).
• If \(D < 0\), the equation has two complex roots.

In our example, we calculated the discriminant as:
\[ D = (-6)^2 - 4(1)(-16) = 36 + 64 = 100 \]
Since \(D = 100\) is greater than zero, it means our quadratic equation has two distinct real roots. This confirms the solutions we found using the Quadratic Formula.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that satisfy the given quadratic equation. There are several methods to solve them, one of which is using the Quadratic Formula as discussed earlier.

Let's summarize the process with steps:

• Step 1: Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
• Step 2: Calculate the discriminant \(D = b^2 - 4ac\).
• Step 3: Use the Quadratic Formula \( x = \frac{-b \, \pm \, \sqrt{D}}{2a} \) and substitute the values of \(a\), \(b\), and \(D\).
• Step 4: Simplify the expression to find the roots.

Consider the example from the exercise: \[ b^2 - 6b - 16 = 0 \]
Follow the steps:

• Identify \(a = 1\), \(b = -6\), \(c = -16\).
• Calculate the discriminant: \(D = 100\).
• Apply the formula: \(x = \frac{6 \, \pm \, \sqrt{100}}{2}\).
• Simplify to get \(x = 8\) and \(x = -2\).

This method guarantees a solution to any solvable quadratic equation.