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Chapter 0: Problem 111

Solve each equation, rounding your answer to four significant digits where necessary. $$ x^{2}-(1+2 x)^{2}=0 $$

Short Answer

Expert verified
The solutions to the equation \(x^2-(1+2x)^{2}=0\) are \(x = 0.2\) and \(x = -1\).

Step by step solution

01

Expand and simplify the equation

: First, we need to expand the expression \((1+2x)^2\) and then simplify the equation. \((1+2x)^2 = (1 + 4x + 4x^2)\) Now, substitute this back into the equation: \(x^2 - (1 + 4x + 4x^2) = 0\)
02

Combine like terms

: Now, combine like terms and simplify the equation further: \(x^2 - 1 - 4x - 4x^2 = 0\) Rearranging the terms, we get: \(4x^2 + x^2 + 4x -1 = 0\) Which simplifies to: \(5x^2 + 4x - 1 = 0\)
03

Solve the quadratic equation

: Now we have a quadratic equation in the form \(ax^2 + bx + c = 0\). We can solve this using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) In our case, \(a = 5\), \(b = 4\), and \(c = -1\). Plugging these values into the formula: \(x = \frac{-4 \pm \sqrt{4^2 - 4\cdot5\cdot(-1)}}{2\cdot5}\)
04

Simplify and find the solutions

: Simplify the expression inside the square root: \(x = \frac{-4 \pm \sqrt{16 + 20}}{10}\) \(x = \frac{-4 \pm \sqrt{36}}{10}\) There will be two different solutions, one using the plus sign and one using the minus sign: 1. Using the plus sign: \(x = \frac{-4 + 6}{10} = \frac{2}{10} = 0.2\) 2. Using the minus sign: \(x = \frac{-4 - 6}{10} = \frac{-10}{10} = -1\) So, the solutions are \(x = 0.2\) and \(x = -1\). Both of them are already rounded to four significant digits.

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