Solve the Quadratic Equation: -81x² + 54x - 9 = 0

Q: What does it mean when the discriminant is zero?

When $ D = 0 $, your quadratic is a perfect square trinomial! This means the parabola just touches the x-axis at exactly one point, giving you one repeated solution instead of two different ones.

Q: Why does the quadratic formula still work here?

The formula $ x = \frac{-b \pm \sqrt{D}}{2a} $ always works! When $ D = 0 $, the $ \pm \sqrt{0} $ part becomes $ \pm 0 $, so you get the same answer twice: one solution.

Q: Could I factor this equation instead?

Yes! First factor out $ -9 $: $ -9(9x^2 - 6x + 1) = 0 $. Then recognize $ 9x^2 - 6x + 1 = (3x - 1)^2 $, giving you $ -9(3x - 1)^2 = 0 $.

Q: How do I check my answer is correct?

Substitute $ x = \frac{1}{3} $ back into the original equation. Calculate each term: $ -81 \cdot \frac{1}{9} = -9 $, $ 54 \cdot \frac{1}{3} = 18 $. Then: $ -9 + 18 - 9 = 0 $ ✓

Q: Why is this called a repeated root?

The term repeated root means the same solution appears twice when you factor completely. Think of $ (3x - 1)^2 = 0 $ as $ (3x - 1)(3x - 1) = 0 $ - both factors give $ x = \frac{1}{3} $!

Quadratic Equations with Zero Discriminant

Solve the following equation:

$-81x^2+54x-9=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Pay attention to the coefficients

00:13 Use the root formula

00:26 Substitute appropriate values according to the given data and solve for X

00:51 Calculate the products and the square

01:17 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:

$-81x^2+54x-9=0$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Identify the coefficients $a = -81$ , $b = 54$ , and $c = -9$ .
Step 2: Calculate the discriminant $D = b^2 - 4ac$ .
Step 3: Apply the quadratic formula to find the solution for $x$ .

Step 1: We have $a = -81$ , $b = 54$ , and $c = -9$ .

Step 2: Calculate the discriminant:

$D = b^2 - 4ac = 54^2 - 4(-81)(-9)$

$= 2916 - 2916 = 0$

Since the discriminant is zero, there is exactly one real solution, indicating a perfect square trinomial.

Step 3: Apply the quadratic formula:

$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-54 \pm \sqrt{0}}{-162}$

$x = \frac{-54}{-162} = \frac{1}{3}$

Therefore, the solution to the problem is $x = \frac{1}{3}$ .

Final Answer

$x=\frac{1}{3}$

Key Points to Remember

Essential concepts to master this topic

Discriminant Rule: When $D = b^2 - 4ac = 0$ , expect one repeated solution
Technique: Factor out GCF first: $-9(9x^2 - 6x + 1) = 0$
Check: Substitute $x = \frac{1}{3}$ : $-81(\frac{1}{9}) + 54(\frac{1}{3}) - 9 = -9 + 18 - 9 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Assuming two different solutions when discriminant is zero
Don't write two separate solutions like $x_1 = \frac{1}{3}, x_2 = \frac{1}{3}$ = confusing notation! When the discriminant equals zero, the quadratic has exactly one solution (a repeated root). Always write it as a single answer: $x = \frac{1}{3}$ .

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

What does it mean when the discriminant is zero?

When $D = 0$ , your quadratic is a perfect square trinomial! This means the parabola just touches the x-axis at exactly one point, giving you one repeated solution instead of two different ones.

Why does the quadratic formula still work here?

The formula $x = \frac{-b \pm \sqrt{D}}{2a}$ always works! When $D = 0$ , the $\pm \sqrt{0}$ part becomes $\pm 0$ , so you get the same answer twice: one solution.

Could I factor this equation instead?

Yes! First factor out $-9$ : $-9(9x^2 - 6x + 1) = 0$ . Then recognize $9x^2 - 6x + 1 = (3x - 1)^2$ , giving you $-9(3x - 1)^2 = 0$ .

How do I check my answer is correct?

Substitute $x = \frac{1}{3}$ back into the original equation. Calculate each term: $-81 \cdot \frac{1}{9} = -9$ , $54 \cdot \frac{1}{3} = 18$ . Then: $-9 + 18 - 9 = 0$ ✓

Why is this called a repeated root?

The term repeated root means the same solution appears twice when you factor completely. Think of $(3x - 1)^2 = 0$ as $(3x - 1)(3x - 1) = 0$ - both factors give $x = \frac{1}{3}$ !

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