Solve the Quadratic Equation with Fractional Terms: x² + 10/9x + 25/81 = 0

Q: How do I recognize a perfect square trinomial?

Look for the pattern $ x^2 + 2dx + d^2 $. The middle term coefficient should be twice the square root of the constant term. Here: $ \sqrt{\frac{25}{81}} = \frac{5}{9} $, and $ 2 \times \frac{5}{9} = \frac{10}{9} $ ✓

Q: Why does a perfect square equal zero give only one solution?

When $ (x + d)^2 = 0 $, the only way a square can equal zero is if the base equals zero: $ x + d = 0 $. This gives us one repeated root rather than two different solutions.

Q: What if the constant term doesn't match d²?

Then it's not a perfect square trinomial! You'll need to use other methods like the quadratic formula or factoring. Always verify that $ d^2 $ equals your constant term before proceeding.

Q: How do I check my answer with fractions?

Substitute $ x = -\frac{5}{9} $ back: $ (-\frac{5}{9})^2 + \frac{10}{9}(-\frac{5}{9}) + \frac{25}{81} = \frac{25}{81} - \frac{50}{81} + \frac{25}{81} = 0 $ ✓

Perfect Square Trinomials with Fractional Coefficients

Solve the following equation:

$x^2+\frac{10}{9}x+\frac{25}{81}=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Identify the coefficients

00:20 Use the roots formula

00:38 Substitute appropriate values according to the given data and solve

00:58 Calculate the squares and products

01:24 A root of 0 is always equal to 0

01:40 When the root equals 0, there will be only one solution to the equation

02:06 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:

$x^2+\frac{10}{9}x+\frac{25}{81}=0$

Step-by-step solution

To solve the given quadratic equation $x^2 + \frac{10}{9}x + \frac{25}{81} = 0$ , we will first check if it can be expressed as a perfect square trinomial.

Notice that a quadratic equation in the form $(x + d)^2 = 0$ expands to:

$(x + d)^2 = x^2 + 2dx + d^2$ .

We observe:

$2d = \frac{10}{9}$ , which gives $d = \frac{10}{18} = \frac{5}{9}$ .
$d^2 = \left(\frac{5}{9}\right)^2 = \frac{25}{81}$ .

This confirms that the original equation can be rewritten as:

$(x + \frac{5}{9})^2 = 0$ .

Solving $(x + \frac{5}{9})^2 = 0$ yields:

$x + \frac{5}{9} = 0$ .

Subtracting $\frac{5}{9}$ from both sides, we get:

$x = -\frac{5}{9}$ .

Therefore, the solution to the equation is $x = -\frac{5}{9}$ , which corresponds to choice id="3".

Final Answer

$x=-\frac{5}{9}$

Key Points to Remember

Essential concepts to master this topic

Recognition: Check if quadratic matches the pattern $x^2 + 2dx + d^2$
Technique: From $\frac{10}{9}x$ , find d: $2d = \frac{10}{9}$ so $d = \frac{5}{9}$
Verify: Check $d^2 = (\frac{5}{9})^2 = \frac{25}{81}$ matches constant term ✓

Common Mistakes

Avoid these frequent errors

Using the quadratic formula unnecessarily
Don't automatically apply the quadratic formula to every quadratic = complicated fractions and extra work! Perfect square trinomials can be solved much faster. Always check if the equation factors as $(x + d)^2 = 0$ first.

Practice Quiz

Test your knowledge with interactive questions

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

FAQ

Everything you need to know about this question

How do I recognize a perfect square trinomial?

Look for the pattern $x^2 + 2dx + d^2$ . The middle term coefficient should be twice the square root of the constant term. Here: $\sqrt{\frac{25}{81}} = \frac{5}{9}$ , and $2 \times \frac{5}{9} = \frac{10}{9}$ ✓

Why does a perfect square equal zero give only one solution?

When $(x + d)^2 = 0$ , the only way a square can equal zero is if the base equals zero: $x + d = 0$ . This gives us one repeated root rather than two different solutions.

What if the constant term doesn't match d²?

Then it's not a perfect square trinomial! You'll need to use other methods like the quadratic formula or factoring. Always verify that $d^2$ equals your constant term before proceeding.

Can I still use the quadratic formula if I want to?

Yes, but it's much more work! The quadratic formula will give $x = \frac{-\frac{10}{9} \pm \sqrt{0}}{2} = -\frac{5}{9}$ . Recognizing perfect squares saves time and reduces calculation errors.

How do I check my answer with fractions?

Substitute $x = -\frac{5}{9}$ back: $(-\frac{5}{9})^2 + \frac{10}{9}(-\frac{5}{9}) + \frac{25}{81} = \frac{25}{81} - \frac{50}{81} + \frac{25}{81} = 0$ ✓

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