Problem 72 - Algebra and Trigonometry

Question

Solve the equation for \(x\). \(b x^{2}+2 x+\frac{1}{b}=0 \quad(b \neq 0)\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{-1}{b}\).

Step 1: Identify the Form of the Equation

The given equation is a quadratic equation in the form \(ax^2 + bx + c = 0\), where \(a = b\), \(b = 2\), and \(c = \frac{1}{b}\). Since it is quadratic, we can use the quadratic formula to find the solutions. The quadratic formula is \(x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\).

Step 2: Assign the Values to A, B, C

For the given equation \(bx^2 + 2x + \frac{1}{b} = 0\), we identify \(A = b\), \(B = 2\), and \(C = \frac{1}{b}\). Substitute these values into the quadratic formula.

Step 3: Substitute Into the Quadratic Formula

Substitute the values for \(A\), \(B\), and \(C\) into the quadratic formula: \[ x = \frac{-2 \pm \sqrt{2^2 - 4(b)(\frac{1}{b})}}{2b} \]. Simplify the expression under the square root.

Step 4: Simplify the Discriminant

Calculate the discriminant \(B^2 - 4AC\) which is \(2^2 - 4(b)(\frac{1}{b}) = 4 - 4 = 0\). Since the discriminant is zero, there will be exactly one real solution.

Step 5: Calculate the Solution

With the discriminant as zero, the quadratic formula simplifies to \(x = \frac{-2}{2b}\). Simplify to find \(x = \frac{-1}{b}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This formula allows you to find the values of \(x\) that satisfy the equation. It is given by:

• \(x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\)

In this formula:

• \(A\) represents the coefficient of \(x^2\)
• \(B\) represents the coefficient of \(x\)
• \(C\) represents the constant term

To use the quadratic formula effectively, start by ensuring your equation is in the standard quadratic form. Identify your coefficients \(A\), \(B\), and \(C\), and substitute them into the formula.
The result will give you values of \(x\) that are the solutions to the equation.
This formula can solve all types of quadratic equations, whether they have one, two or no real solutions.

Discriminant

The discriminant is a key part of the quadratic formula and is crucial for understanding the nature of the solutions to a quadratic equation. It is found under the square root symbol in the quadratic formula, expressed as \(B^2 - 4AC\).
This value helps determine how many real solutions the quadratic equation has:

• If the discriminant is positive, the equation has two distinct real solutions.
• If the discriminant is zero, the equation has exactly one real solution (a repeated root).
• If the discriminant is negative, the equation has no real solutions (but two complex ones).

In the original exercise, the discriminant was calculated as zero,

• \(2^2 - 4(b)(\frac{1}{b}) = 4 - 4 = 0\)

indicating one real solution exists.Understanding the discriminant not only aids in finding solutions but also in predicting the type of solutions you might encounter.

Real Solution

Real solutions of a quadratic equation are the values of \(x\) where the quadratic expression becomes zero on the real number line. These solutions intersect the \(x\)-axis on a graph of the quadratic function.
For a quadratic equation to have real solutions,

• its discriminant \(B^2 - 4AC\) must be zero or positive.

In the scenario where the discriminant is zero, as in our original exercise, the quadratic equation has only one real solution.
Specifically for the exercise, once the quadratic formula was applied with \(A = b\), \(B = 2\), and \(C = \frac{1}{b}\), the calculation revealed that the single real solution was \(x = \frac{-1}{b}\).
This means at that value of \(x\), the graph of the quadratic equation touches the \(x\)-axis.Identifying and calculating real solutions is essential for problems involving physical phenomena, optimization, and other applications in mathematics where real-world interpretations of solutions are needed.