Question

Solve the equation by the method of your choice.
StartFraction 1 Over x EndFraction plus StartFraction 1 Over x plus 2 EndFraction equals one seventh

Answer 1

To solve the equation

\[
\frac{1}{x} + \frac{1}{x+2} = \frac{1}{7},
\]

we can begin by finding a common denominator for the left-hand side. The common denominator for the fractions \(\frac{1}{x}\) and \(\frac{1}{x+2}\) is \(x(x+2)\). Thus, we can rewrite the equation:

\[
\frac{(x+2) + x}{x(x+2)} = \frac{1}{7}.
\]

This simplifies the left-hand side:

\[
\frac{2x + 2}{x(x+2)} = \frac{1}{7}.
\]

Next, we can cross-multiply to eliminate the fractions:

\[
7(2x + 2) = x(x + 2).
\]

Expanding both sides gives:

\[
14x + 14 = x^2 + 2x.
\]

Rearranging this into a standard quadratic equation form, we subtract \(14x + 14\) from both sides:

\[
x^2 + 2x - 14x - 14 = 0,
\]

which simplifies to:

\[
x^2 - 12x - 14 = 0.
\]

Now, we will apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -12\), and \(c = -14\):

1. Calculate the discriminant:

\[
b^2 - 4ac = (-12)^2 - 4(1)(-14) = 144 + 56 = 200.
\]

2. Apply the quadratic formula:

\[
x = \frac{-(-12) \pm \sqrt{200}}{2(1)} = \frac{12 \pm \sqrt{200}}{2}.
\]

3. Simplify \(\sqrt{200}\):

\[
\sqrt{200} = \sqrt{100 \cdot 2} = 10\sqrt{2}.
\]

4. Substitute this back into the quadratic formula:

\[
x = \frac{12 \pm 10\sqrt{2}}{2} = 6 \pm 5\sqrt{2}.
\]

Thus, the solution for \(x\) is

\[
x = 6 + 5\sqrt{2} \quad \text{and} \quad x = 6 - 5\sqrt{2}.
\]

Both values are valid solutions for the original equation.