Solve the Quadratic Fraction Equation: x^2/9 + 2/9x + 1/9 = 0

Q: Why do we multiply by 9 instead of solving with fractions?

Multiplying by 9 clears all the fractions at once, making the equation much simpler to work with. It's like getting rid of the denominators so you can focus on solving $ x^2 + 2x + 1 = 0 $ instead.

Q: How do I recognize this is a perfect square trinomial?

Look for the pattern $ a^2 + 2ab + b^2 = (a + b)^2 $. Here, $ x^2 + 2x + 1 $ fits with a = x and b = 1, so it factors as $ (x + 1)^2 $.

Q: Why is there only one solution when quadratics usually have two?

This quadratic has a repeated root! When $ (x + 1)^2 = 0 $, both solutions are the same: x = -1. It's like the parabola just touches the x-axis at one point.

Q: Can I use the quadratic formula instead of factoring?

Absolutely! The quadratic formula will give you $ x = \frac{-2 \pm \sqrt{4-4}}{2} = \frac{-2 \pm 0}{2} = -1 $. Factoring is just faster when you recognize the pattern.

Q: What if I made an error clearing the fractions?

Double-check by ensuring every term gets multiplied by 9. The most common mistake is forgetting to multiply the constant term $ \frac{1}{9} $ by 9 to get 1.

Quadratic Equations with Perfect Square Factoring

Solve the following equation:

$\frac{x^2}{9}+\frac{2}{9}x+\frac{1}{9}=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Multiply by 9 to eliminate fractions

00:17 Identify the coefficients

00:21 Use the roots formula

00:41 Substitute appropriate values according to the given data and solve

01:02 Calculate the square and products

01:17 A root of 0 is always equal to 0

01:21 When the root equals 0, there will be only one solution to the equation

01:43 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$\frac{x^2}{9}+\frac{2}{9}x+\frac{1}{9}=0$

Step-by-step solution

To solve this equation, we shall proceed with these steps:

Step 1: Clear the fractions. Multiply each term of the equation by 9 to simplify:
$9 \left(\frac{x^2}{9}\right) + 9 \left(\frac{2}{9}x\right) + 9 \left(\frac{1}{9}\right) = 9 \times 0$
This simplifies down to $x^2 + 2x + 1 = 0$ .
Step 2: Recognize $x^2 + 2x + 1 = 0$ as a quadratic equation, which can be factored as:
$(x + 1)^2 = 0$ .
Step 3: Solving $(x + 1)^2 = 0$ gives $x + 1 = 0$ , thus $x = -1$ .

Therefore, the solution to the equation is $x = -1$ .

The correct choice that corresponds to this solution from the provided options is $x = -1$ .

Final Answer

$x=-1$

Key Points to Remember

Essential concepts to master this topic

Simplification: Multiply all terms by 9 to clear fractions first
Factoring: Recognize $x^2 + 2x + 1 = (x + 1)^2$ perfect square trinomial
Verification: Substitute x = -1: $\frac{(-1)^2}{9} + \frac{2(-1)}{9} + \frac{1}{9} = 0$ ✓

Common Mistakes

Avoid these frequent errors

Not multiplying all terms by the same number to clear fractions
Don't multiply only some terms by 9 and leave others as fractions = mixed equation that's harder to solve! This creates confusion and calculation errors. Always multiply every single term by the same number to clear all fractions at once.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ 2x^2-10x-12=0 $$

FAQ

Everything you need to know about this question

Why do we multiply by 9 instead of solving with fractions?

Multiplying by 9 clears all the fractions at once, making the equation much simpler to work with. It's like getting rid of the denominators so you can focus on solving $x^2 + 2x + 1 = 0$ instead.

How do I recognize this is a perfect square trinomial?

Look for the pattern $a^2 + 2ab + b^2 = (a + b)^2$ . Here, $x^2 + 2x + 1$ fits with a = x and b = 1, so it factors as $(x + 1)^2$ .

Why is there only one solution when quadratics usually have two?

This quadratic has a repeated root! When $(x + 1)^2 = 0$ , both solutions are the same: x = -1. It's like the parabola just touches the x-axis at one point.

Can I use the quadratic formula instead of factoring?

Absolutely! The quadratic formula will give you $x = \frac{-2 \pm \sqrt{4-4}}{2} = \frac{-2 \pm 0}{2} = -1$ . Factoring is just faster when you recognize the pattern.

What if I made an error clearing the fractions?

Double-check by ensuring every term gets multiplied by 9. The most common mistake is forgetting to multiply the constant term $\frac{1}{9}$ by 9 to get 1.

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