Evaluate x²+9/x² Given x+3/x=5: No Solving Required

Q: Why can't I just solve for x and then substitute?

While possible, it's much more complicated! Solving $ x+\frac{3}{x}=5 $ gives you a quadratic equation. The squaring method is faster and avoids potential calculation errors.

Q: How does squaring help me get from first powers to second powers?

When you square $ (x+\frac{3}{x}) $, you get $ x^2 + 2 \cdot 3 + \frac{9}{x^2} $. The middle term becomes a constant, leaving you with exactly what you need!

Q: What if the middle term doesn't cancel out nicely?

In problems like this, the middle term always simplifies because $ x \cdot \frac{3}{x} = 3 $. This is why these problems work - the variable cancels in the cross term!

Q: Do I need to expand the binomial formula every time?

For practice, yes! Write out $ (a+b)^2 = a^2 + 2ab + b^2 $ each time. This helps you see the pattern and avoid missing the middle term.

Q: How do I know this method will work for other similar problems?

Look for the pattern: if you have $ x + \frac{k}{x} $ and need $ x^2 + \frac{k^2}{x^2} $, squaring always works because the middle term becomes $ 2k $ (a constant).

Q: What's the final step after expanding?

Isolate the target expression! From $ x^2 + 6 + \frac{9}{x^2} = 25 $, subtract 6 from both sides to get $ x^2 + \frac{9}{x^2} = 19 $.

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

^{The given equation is:}

$x+\frac{3}{x}=5$

Calculate, without solving the equation for x

the value of the expression

$x^2+\frac{9}{x^2}=\text{?}$

2

Step-by-step solution

Our goal is to calculate the value of the following expression:

$x^2+\frac{9}{x^2}=\text{?}$

based on the given equation:

$x+\frac{3}{x}=5$

However without solving it for x,

Note that while the given equation deals with terms raised to the first power only,

in the expression we want to calculate - there are terms raised to the second power.

Therefore we need to square the expression on the left side of the given equation.

We can do this by using the shortened multiplication formula for binomial square:

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

We'll proceed to square both sides of the given equation, later on we'll discuss what happens in the given mathematical structure when a term and its proportional inverse are added:

$x+\frac{3}{x}=5 \hspace{6pt}\text{/}()^2\\ (x+\frac{3}{x})^2=5^2\\ \downarrow\\ x^2+2\cdot \textcolor{blue}{x\cdot \frac{3}{x}}+ \frac{3^2}{x^2}=25\\ \downarrow\\ x^2+2\cdot \textcolor{blue}{3}+ \frac{9}{x^2}=25\\$ For now take note that the "mixed" term in the shortened multiplication formula ( $2ab$ ) gives us - as a result of squaring the mathematical structure in question - a free number. This signifies that it's not dependent on the variable x, since it involves multiplication between an expression with a variable and its proportional inverse.

This fact actually allows us to isolate the desired expression from the equation we obtain. We can subsequently determine its value (which is not dependent on the variable) even without knowing the value of the unknown (or unknowns) that solves the equation:

$x^2+2\cdot \textcolor{blue}{3}+ \frac{9}{x^2}=25\\ x^2+6+ \frac{9}{x^2}=25\\ \boxed{x^2+\frac{9}{x^2}=19}$

Therefore the correct answer is answer C.

3

Final Answer

$19$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Use squaring when given terms have lower powers
Technique: Square both sides: $(x+\frac{3}{x})^2 = (5)^2$
Check: Middle term simplifies to constant: $2 \cdot x \cdot \frac{3}{x} = 6$ ✓

Common Mistakes

Avoid these frequent errors

Attempting to solve for x first
Don't try to solve $x+\frac{3}{x}=5$ for x values = leads to quadratic complications! This creates unnecessary work and potential errors. Always use algebraic manipulation to transform the given expression directly into the target form.

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 substitute?

+

While possible, it's much more complicated! Solving $x+\frac{3}{x}=5$ gives you a quadratic equation. The squaring method is faster and avoids potential calculation errors.

How does squaring help me get from first powers to second powers?

+

When you square $(x+\frac{3}{x})$ , you get $x^2 + 2 \cdot 3 + \frac{9}{x^2}$ . The middle term becomes a constant, leaving you with exactly what you need!

What if the middle term doesn't cancel out nicely?

+

In problems like this, the middle term always simplifies because $x \cdot \frac{3}{x} = 3$ . This is why these problems work - the variable cancels in the cross term!

Do I need to expand the binomial formula every time?

+

For practice, yes! Write out $(a+b)^2 = a^2 + 2ab + b^2$ each time. This helps you see the pattern and avoid missing the middle term.

How do I know this method will work for other similar problems?

+

Look for the pattern: if you have $x + \frac{k}{x}$ and need $x^2 + \frac{k^2}{x^2}$ , squaring always works because the middle term becomes $2k$ (a constant).

What's the final step after expanding?

+

Isolate the target expression! From $x^2 + 6 + \frac{9}{x^2} = 25$ , subtract 6 from both sides to get $x^2 + \frac{9}{x^2} = 19$ .

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Evaluate x²+9/x² Given x+3/x=5: No Solving Required

Algebraic Manipulation with Squaring Technique

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