Calculate x² + 1/x² Given x + 1/x = 4: Indirect Solution Method

Q: Why can't I just solve for x and then calculate x² + 1/x²?

You could, but it's much harder! Solving $ x + \frac{1}{x} = 4 $ gives you a quadratic equation with messy solutions. The indirect method is elegant and avoids complex calculations.

Q: How does squaring both sides help me find x² + 1/x²?

When you square $ (x + \frac{1}{x})^2 $, you get $ x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2} $. The middle term $ x \cdot \frac{1}{x} = 1 $, so you can easily isolate $ x^2 + \frac{1}{x^2} $!

Q: What if x + 1/x was a different number, like 3?

The same method works! If $ x + \frac{1}{x} = 3 $, then $ (x + \frac{1}{x})^2 = 9 $, giving $ x^2 + \frac{1}{x^2} = 9 - 2 = 7 $. The pattern is always square minus 2.

Q: Why does x · (1/x) always equal 1?

This is a fundamental property of reciprocals! Any number multiplied by its reciprocal equals 1. Think: $ \frac{3}{1} \times \frac{1}{3} = \frac{3}{3} = 1 $. This works for any non-zero value of x.

Q: Is 14 the only possible answer for x² + 1/x²?

Yes! Even though the original equation $ x + \frac{1}{x} = 4 $ has two solutions for x, both solutions give the same value for $ x^2 + \frac{1}{x^2} $. That's the beauty of this method!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

The given equation is:

$x+\frac{1}{x}=4$

Calculate, without solving the equation for x,

The value of the expression:

$x^2+\frac{1}{x^2}=\text{?}$

2

Step-by-step solution

We want to calculate the value of the expression:

$x^2+\frac{1}{x^2}=\text{?}$

based on the given equation:

$x+\frac{1}{x}=4$

but without solving it for x,

For this, let's first note that while the given equation deals with terms with first power only,

in the expression whose value we want to calculate - there are terms with second power only,

therefore we understand that apparently we need to square the expression on the left side of the given equation,

We'll remember of course the shortened multiplication formula for a binomial square:

$(a\pm b)^2=a^2\pm2ab+b^2$

and we'll square both sides of the given equation, later we'll emphasize something worth noting that happens in the given mathematical structure in the expression in question (the mathematical structure where a term and its inverse are added):

$x+\frac{1}{x}=4 \hspace{6pt}\text{/}()^2\\ (x+\frac{1}{x})^2=4^2\\ \downarrow\\ x^2+2\cdot \textcolor{blue}{x\cdot \frac{1}{x}}+ \frac{1}{x^2}=16\\ \downarrow\\ x^2+2\cdot \textcolor{blue}{1}+ \frac{1}{x^2}=16\\$ Let's now notice that the "mixed" term in the shortened multiplication formula ( $2ab$ ) gives us - from squaring the mathematical structure in question - a free number, meaning - it's not dependent on the variable x, since it's a multiplication between an expression and its inverse,

This fact actually allows us to isolate the desired expression from the equation we get and find its value (which is not dependent on the variable) even without knowing the value of the unknown (or unknowns) that solves the equation:

$x^2+2\cdot \textcolor{blue}{1}+ \frac{1}{x^2}=16\\ x^2+2+ \frac{1}{x^2}=16\\ \boxed{x^2+\frac{1}{x^2}=14}$

Therefore the correct answer is answer D.

3

Final Answer

$14$

Key Points to Remember

Essential concepts to master this topic

Identity Rule: Square both sides to create relationship between given and desired expressions
Technique: Apply $(a+b)^2 = a^2 + 2ab + b^2$ where mixed term simplifies
Check: Verify $x \cdot \frac{1}{x} = 1$ makes mixed term equal 2 ✓

Common Mistakes

Avoid these frequent errors

Trying to solve for x first instead of using algebraic manipulation
Don't solve $x + \frac{1}{x} = 4$ for x values = complex quadratic with messy solutions! This defeats the purpose of the indirect method. Always square both sides directly to create the connection between given and desired expressions.

Practice Quiz

Test your knowledge with interactive questions

_{Look at the following equation:}

_{$$ 16x^2+24x-40=0 $$}

_{Using the method of completing the square and without solving the equation for X, calculate the value of the following expression:}

_{$12x+9=\text{?}$}

FAQ

Everything you need to know about this question

Why can't I just solve for x and then calculate x² + 1/x²?

+

You could, but it's much harder! Solving $x + \frac{1}{x} = 4$ gives you a quadratic equation with messy solutions. The indirect method is elegant and avoids complex calculations.

How does squaring both sides help me find x² + 1/x²?

+

When you square $(x + \frac{1}{x})^2$ , you get $x^2 + 2 \cdot x \cdot \frac{1}{x} + \frac{1}{x^2}$ . The middle term $x \cdot \frac{1}{x} = 1$ , so you can easily isolate $x^2 + \frac{1}{x^2}$ !

What if x + 1/x was a different number, like 3?

+

The same method works! If $x + \frac{1}{x} = 3$ , then $(x + \frac{1}{x})^2 = 9$ , giving $x^2 + \frac{1}{x^2} = 9 - 2 = 7$ . The pattern is always square minus 2.

Why does x · (1/x) always equal 1?

+

This is a fundamental property of reciprocals! Any number multiplied by its reciprocal equals 1. Think: $\frac{3}{1} \times \frac{1}{3} = \frac{3}{3} = 1$ . This works for any non-zero value of x.

Is 14 the only possible answer for x² + 1/x²?

+

Yes! Even though the original equation $x + \frac{1}{x} = 4$ has two solutions for x, both solutions give the same value for $x^2 + \frac{1}{x^2}$ . That's the beauty of this method!

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