Problem 76 - 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 to the equation is \( x = -\frac{1}{b} \).

Step 1: Identify the Quadratic Equation

The given equation is a quadratic in the standard form \( ax^2 + bx + c = 0 \) where \( a = b \), \( b = 2 \), and \( c = \frac{1}{b} \).

Step 2: Recognize the Quadratic Formula

Recall the quadratic formula for solving equations of the form \( ax^2 + bx + c = 0 \): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 3: Substitute Coefficients into Quadratic Formula

Substitute \( a = b \), \( b = 2 \), and \( c = \frac{1}{b} \) into the quadratic formula:\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot b \cdot \frac{1}{b}}}{2b} \]

Step 4: Simplify Inside the Square Root

Calculate inside the square root:\[ 2^2 - 4b \cdot \frac{1}{b} = 4 - 4 = 0 \] So, the expression under the square root simplifies to 0.

Step 5: Evaluate the Quadratic Formula

The expression now becomes:\[ x = \frac{-2 \pm \sqrt{0}}{2b} = \frac{-2}{2b} \]

Step 6: Simplify the Solution

Simplify \( \frac{-2}{2b} \) to find the value of \( x \):\[ x = -\frac{1}{b} \]

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 provides a straightforward way to find the values of \( x \) that satisfy the equation. The quadratic formula is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula works for any quadratic equation as long as we know the coefficients \( a \), \( b \), and \( c \). Here, \( a \) is the coefficient of \( x^2 \), \( b \) is the coefficient of \( x \), and \( c \) is the constant term.

• The "+/-" symbol indicates that there may be two possible solutions for \( x \).
• The term \( b^2 - 4ac \) under the square root is known as the discriminant.
• The discriminant determines the nature of the roots of the equation.

Understanding and applying this formula is central in algebra, especially when solutions are not readily apparent by simple factorization.

Solving Quadratic Equations

Solving quadratic equations involves finding all possible values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). Depending on the discriminant \( b^2 - 4ac \), you might find:

• Two distinct real roots, if the discriminant is positive.
• One real root, if the discriminant is zero.
• No real roots (but two complex roots), if the discriminant is negative.

In the given problem, after substituting the values into the quadratic formula, we found that the discriminant equaled zero. This means that the quadratic equation has exactly one real root.
Following the steps:

• Identify the coefficients from the standard form.
• Substitute these coefficients into the quadratic formula.
• Simplify to find the value(s) of \( x \).

Applying these straightforward steps helps solve any quadratic equation, ensuring that you accurately find all potential solutions.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions to reveal the values of unknowns. When solving a quadratic equation, algebraic manipulation involves careful substitution and simplification of expressions:

In our example, we first identified the equation in standard quadratic form; then, we defined \( a \), \( b \), and \( c \). Upon substituting these into the quadratic formula:\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot b \cdot \frac{1}{b}}}{2b} \]The inside of the square root, known as the discriminant, was simplified:

• \( 2^2 = 4 \)
• \( 4 \cdot b \cdot \frac{1}{b} = 4 \), leading to \( 4 - 4 = 0 \)

Since the discriminant was zero, it simplified the calculation of \( x \) to:\[ x = \frac{-2}{2b} \]And further simplifying gave:\[ x = -\frac{1}{b} \]
These manipulations are crucial, as they not only simplify complex expressions but also help in understanding the relationship between the components of the quadratic equation.