Question

1. the sum of a number, x, and its reciprocal, is $\frac{29}{10}$. form an equation and find the original number.

Explanation:

Step1: Set up the equation

The reciprocal of \(x\) is \(\frac{1}{x}\). The sum of \(x\) and its reciprocal is \(x+\frac{1}{x}\), and we know it equals \(\frac{29}{10}\). So the equation is \(x + \frac{1}{x}=\frac{29}{10}\).

Step2: Multiply through by \(10x\) to clear fractions

\[10x\times x+10x\times\frac{1}{x}=10x\times\frac{29}{10}\]
\[10x^{2}+ 10 = 29x\]

Step3: Rearrange to quadratic - form

\[10x^{2}-29x + 10=0\]

Step4: Factor the quadratic equation

We need to find two numbers that multiply to \(10\times10 = 100\) and add up to \(-29\). The numbers are \(-25\) and \(-4\).
\[10x^{2}-25x-4x + 10 = 0\]
\[5x(2x - 5)-2(2x - 5)=0\]
\((5x - 2)(2x - 5)=0\)

Step5: Solve for \(x\)

If \(5x-2 = 0\), then \(5x=2\), and \(x=\frac{2}{5}\).
If \(2x - 5=0\), then \(2x=5\), and \(x=\frac{5}{2}\).

Answer:

\(x=\frac{2}{5}\) or \(x=\frac{5}{2}\)