3/x + 1 = 2x
with step by step solution
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Answer:
To solve the equation \( \frac{3}{x} + 1 = 2x \), follow these steps:
Step 1: Subtract 1 from both sides of the equation to isolate the fraction:
\(\frac{3}{x} = 2x - 1\)
Step 2: Get rid of the fraction by multiplying both sides of the equation by \(x\) to clear the denominator:
\(3 = 2x^2 - x\)
Step 3: Rearrange the equation in standard quadratic form, which is \(ax^2 + bx + c = 0\):
\(2x^2 - x - 3 = 0\)
Step 4: To solve this quadratic equation, you can use the quadratic formula:
\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
In this case, \(a = 2\), \(b = -1\), and \(c = -3\).
Step 5: Plug the values into the formula and solve for \(x\):
\(x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot (-3)}}{2 \cdot 2}\)
\(x = \frac{1 \pm \sqrt{1 + 24}}{4}\)
\(x = \frac{1 \pm \sqrt{25}}{4}\)
\(x = \frac{1 \pm 5}{4}\)
Step 6: Solve for both possible values of \(x\):
a) \(x = \frac{1 + 5}{4} = \frac{6}{4} = \frac{3}{2}\)
b) \(x = \frac{1 - 5}{4} = \frac{-4}{4} = -1\)
So, the solutions to the equation \( \frac{3}{x} + 1 = 2x \) are \(x = \frac{3}{2}\) and \(x = -1\).