Question

solve for d. \\(\frac{-d - 6}{5} = \frac{1}{d}\\) there may be 1 or 2 solutions. \\(d = \square\\) or \\(d = \square\\)

Explanation:

Step1: Cross - multiply to eliminate fractions

Given the equation \(\frac{-d - 6}{5}=\frac{1}{d}\), cross - multiplying (since \(d
eq0\) and \(5
eq0\)) gives us \(d(-d - 6)=5\times1\).

Step2: Expand and rearrange into standard quadratic form

Expanding the left - hand side: \(-d^{2}-6d = 5\).
Rearranging all terms to one side to get a quadratic equation: \(d^{2}+6d + 5 = 0\) (we multiply both sides by \(- 1\) to make the coefficient of \(d^{2}\) positive).

Step3: Factor the quadratic equation

We factor \(d^{2}+6d + 5\). We need two numbers that multiply to \(5\) and add up to \(6\). The numbers are \(1\) and \(5\). So, \(d^{2}+6d + 5=(d + 1)(d+5)=0\).

Step4: Solve for \(d\)

Using the zero - product property, if \((d + 1)(d + 5)=0\), then either \(d+1 = 0\) or \(d + 5=0\).

• If \(d+1=0\), then \(d=-1\).
• If \(d + 5=0\), then \(d=-5\).

We also need to check if these solutions are valid in the original equation (since we had a fraction with \(d\) in the denominator, \(d
eq0\), and both \(-1\) and \(-5\) are not equal to \(0\)).

Answer:

\(d=-1\) or \(d = - 5\)