Problem 69

Question

Solve each equation. Check your answers. $\frac{1}{3 x+1}=\frac{1}{x^{2}-3}$

Step-by-Step Solution

Verified

Answer

The solutions to the given equation are $x = 4$ and $x = -1$

1Step 1: Clear the equation of fractions

Multiply through by $(3x + 1)(x^2 - 3)$ on both sides which is the least common multiple (LCM) of the denominators: $(3x + 1) \cdot (x^2 - 3) \cdot \frac{1}{3x + 1} = (3x + 1) \cdot (x^2 - 3) \cdot \frac{1}{x^2 - 3}$. This simplifies to: $x^2 - 3 = 3x + 1$

2Step 2: Rearrange the equation

Rearrange the above equation to one side, to get the equation in a standard form. This gives: $x^2 - 3x - 4 = 0$

3Step 3: Factor the quadratic

Factor the quadratic equation to find the roots. The factored form of the equation is $(x - 4) * (x + 1) = 0$

4Step 4: Solve for x

Setting each factor equal to zero gives the solutions $x = 4, x = -1$

5Step 5: Check the solutions

Substitute each solution back into the original equation to verify the validity. Substituting $x = 4$ gives $1/13 = 1/13$ which is correct. Likewise, substituting $x = -1$ gives $-(1/2) = -(1/2)$, which is also correct. Therefore, $x = 4, -1$ are the correct solutions to the given equation.

Key Concepts

Understanding the Least Common MultipleMastering Factoring QuadraticsChecking Solutions for AccuracyRearranging Equations Effectively

Understanding the Least Common Multiple

When solving rational equations, finding the least common multiple (LCM) is key. The LCM of denominators helps clear pesky fractions. In our equation,

Denominators are: $3x + 1$ and $x^2 - 3$.
The LCM of denominators is: $(3x + 1)(x^2 - 3)$.

By multiplying every term by this LCM, fractions vanish, simplifying our life by giving us a straightforward polynomial equation to tackle.

Mastering Factoring Quadratics

Factoring quadratics can feel like an art form, but it’s crucial for solving equations. Once we have our polynomial $x^2 - 3x - 4 = 0$, we:

Look for two numbers that multiply to $-4$ (constant term) and add up to $-3$ (coefficient of $x$).
These numbers are $-4$ and $1$.

Thus, the equation factors to $(x - 4)(x + 1) = 0$. Each factor gives a solution: $x = 4$ and $x = -1$. This method converts a quadratic equation into simpler linear equations.

Checking Solutions for Accuracy

Always check solutions to ensure they're correct! Substitute back into the original equation:

For $x = 4$: Substitute into $\frac{1}{3x+1} = \frac{1}{x^2-3}$, resulting in $\frac{1}{13} = \frac{1}{13}$, confirming $x = 4$ is valid.
For $x = -1$: Substitute to see $-\frac{1}{2} = -\frac{1}{2}$, confirming $x = -1$ is also valid.

Checking step helps you catch errors and confirm that solutions fit the original equation setups.