Problem 28 - Intermediate Algebra

Question

Solve equation. \(\frac{b+1}{2}-\frac{3}{2}=\frac{4}{b}\)

Step-by-Step Solution

Answer

The solution is \( b = 4 \).

Step 1: Simplify the Left Side

Starting with the equation \( \frac{b+1}{2} - \frac{3}{2} = \frac{4}{b} \), let's simplify the left side. Subtract the fractions: \( \frac{b+1-3}{2} \). Further, simplify it to: \( \frac{b-2}{2} \). Now, the equation looks like: \( \frac{b-2}{2} = \frac{4}{b} \).

Step 2: Cross Multiply

Cross multiply the simplified equation: \( (b-2) \cdot b = 4 \cdot 2 \). Expand this to get: \( b^2 - 2b = 8 \).

Step 3: Rearrange into a Standard Quadratic Equation

Bring all terms to one side to form a standard quadratic equation: \( b^2 - 2b - 8 = 0 \).

Step 4: Use the Quadratic Formula

For the quadratic equation \( b^2 - 2b - 8 = 0 \), use the quadratic formula: \( b = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \), where \( A = 1, B = -2, C = -8 \).

Step 5: Calculate the Discriminant

Calculate the discriminant: \( B^2 - 4AC = (-2)^2 - 4 \cdot 1 \cdot (-8) = 4 + 32 = 36 \).

Step 6: Solve the Quadratic Equation

Now substitute back into the quadratic formula: \( b = \frac{-(-2) \pm \sqrt{36}}{2 \cdot 1} \) which simplifies to \( b = \frac{2 \pm 6}{2} \). Determine the two possible solutions: \( b = \frac{8}{2} = 4 \) and \( b = \frac{-4}{2} = -2 \).

Step 7: Verify Potential Solutions

Verify which solutions are valid by substituting back into the original equation. The solution \( b = 4 \) is valid as it satisfies the original equation, but \( b = -2 \) does not because it yields division by zero in the original equation.

Key Concepts

Cross Multiplication in Equations

Cross multiplication is a valuable technique used to solve equations that involve fractions. It allows you to eliminate the fractions, making the equation easier to handle.
Begin by rewriting the equation so that each side is a single fraction. In this case: \[ \frac{b-2}{2} = \frac{4}{b} \] Here, cross multiplication involves multiplying the numerator of one side by the denominator of the other and vice versa.
This converts the fraught fractional landscape into a more straightforward equation: \[ (b-2) \cdot b = 2 \cdot 4 \]

• This results in: \( b^2 - 2b = 8 \)

With fractions gone, you're a step closer to an elegant quadratic equation. Such simplification is very helpful for tackling equations that seem intimidating at first glance.
Mastering cross multiplication can effectively demystify many problems by transforming complex equations into simpler ones.

Understanding the Quadratic Formula

The Quadratic Formula is a powerful tool in algebra used to find the solutions of quadratic equations, which are polynomials of degree two. These take the form: \[ ax^2 + bx + c = 0 \] With coefficients \( a \), \( b \), and \( c \). The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula can solve any quadratic equation, provided you know \( a \), \( b \), and \( c \).

• For the example equation \( b^2 - 2b - 8 = 0 \), the values are \( A = 1 \), \( B = -2 \), \( C = -8 \).
• Inserting these values into the formula helps solve for \( b \).

Using the quadratic formula always provides solutions, even for equations that cannot be factored easily. This makes it indispensable when confronting quadratic equations in various mathematical settings.

Discriminant: Deciding the Nature of Solutions

The term inside the square root of the Quadratic Formula, known as the discriminant, is crucial in determining the nature of solutions a quadratic equation may have.
It is expressed as: \[ b^2 - 4ac \]

• If the discriminant is positive, the equation will have two distinct real roots.
• If it's zero, there will be exactly one real root.
• A negative discriminant indicates two complex roots.

In the problem, the discriminant calculated was \( 36 \). This positive value implies that our equation has two distinct real solutions. Understanding the discriminant is crucial as it offers insight into the equation's behavior even before solving it completely. It sets the expectations for the roots you'll handle along your solving journey.

Verifying Solutions to Ensure Accuracy

Verification is an essential step in solving equations to ensure that the derived solutions satisfy the original condition.
For quadratic equations, you might arrive at multiple solutions; not all may be valid in the context of the problem.

• Substitute each solution back into the original equation to check its validity.
• In the example, check both solutions: \( b = 4 \) and \( b = -2 \).

The solution \( b = 4 \) satisfies the equation, but \( b = -2 \) results in division by zero. Insight like this shows why verification is a crucial step. It saves from assumptions and ensures correctness through rigorous evaluation.
This final step not only boosts confidence in your solutions but also solidifies your understanding of the mathematical concepts involved.