Problem 131 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. To solve \(x-9 \sqrt{x}+14=0,\) we let \(\sqrt{u}=x\)

Step-by-Step Solution

Answer

The statement is false. The correct substitution is \(u = \sqrt{x}\) and the corrected equation should be \(u^2-9u+14=0\). Solving this equation gives us the solutions for \(x\) as 4 and 49.

Step 1: Analyze the statement

The given equation is \(x-9 \sqrt{x}+14=0\), and it's suggested that we let \(\sqrt{u}=x\). However, if we want to eradicate the square root from the equation, the correct substitution would be to let \(u\) substitute \(\sqrt{x}\), not \(\sqrt{u}\). So, let \(u = \sqrt{x}\) or \(x = u^2\). Therefore from this we can conclude that the original statement is false.

Step 2: Substitute the correct variable

We should substitute \(x = u^2\) into the given equation to get rid of the square root. So after substitution, the equation would be: \(u^2-9u+14=0\).

Step 3: Validate the work

Solving the quadratic equation \(u^2-9u+14=0\) will give the values of \(u\). Using the quadratic formula, i.e., \(u = \frac{-b\pm \sqrt{b^2-4ac}}{2a}\) yields the values \(u = 2\) and \(u = 7\). Keep in mind that \(u\) is the square root of \(x\), so \(x = u^2\), and substituting \(2^2\) and \(7^2\) into \(x\) will yield 4 and 49 respectively. These are valid solutions to the equation.