Solving for X in the Equation: x^2 + x + 1/4 = 0

Q: How do I recognize a perfect square trinomial?

Look for the pattern $ x^2 + 2bx + b^2 $. In this case, we have $ x^2 + x + \frac{1}{4} $, where b = 1/2 because $ 2 \cdot \frac{1}{2} = 1 $ and $ (\frac{1}{2})^2 = \frac{1}{4} $.

Q: What if the equation doesn't form a perfect square?

Then you'll need to use other methods like the quadratic formula or factoring. But always check for perfect squares first - they're the easiest to solve!

Q: Why does this equation have only one solution?

When $ (x + \frac{1}{2})^2 = 0 $, the only way a square can equal zero is if the expression inside equals zero. This gives us a repeated root or double root at $ x = -\frac{1}{2} $.

Q: Can I solve this using the quadratic formula instead?

Yes, but it's more work! Using $ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $ with a=1, b=1, c=1/4 gives the same answer, but recognizing the perfect square is much faster.

Q: How do I check my answer?

Substitute $ x = -\frac{1}{2} $ into the original equation: $ (-\frac{1}{2})^2 + (-\frac{1}{2}) + \frac{1}{4} = \frac{1}{4} - \frac{1}{2} + \frac{1}{4} = 0 $ ✓

Quadratic Equations with Perfect Square Form

Solve the following equation:

$x^2+x+\frac{1}{4}=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 Identify the coefficients

00:16 Use the roots formula

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

00:57 Calculate the squares and products

01:07 A root of 0 is always equal to 0

01:19 When the root equals 0, there will be only one solution to the equation

01:42 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:

$x^2+x+\frac{1}{4}=0$

Step-by-step solution

To solve the quadratic equation $x^2 + x + \frac{1}{4} = 0$ , we will use the method of completing the square:

Step 1: Rewrite the equation focusing on forming a perfect square.
Step 2: The expression $x^2 + x$ can be transformed into a perfect square.
Step 3: Note that $(x + \frac{1}{2})^2 = x^2 + x + \frac{1}{4}$ .
Step 4: Substitute into the equation: $(x + \frac{1}{2})^2 = 0$ .
Step 5: Solve for $x$ : If $(x + \frac{1}{2})^2 = 0$ , then $x + \frac{1}{2} = 0$ .
Step 6: Simplifying the equation, we find $x = -\frac{1}{2}$ .

This means the solution to the quadratic equation is $x = -\frac{1}{2}$ .

Thus, the correct answer is $x = -\frac{1}{2}$ .

Final Answer

$x=-\frac{1}{2}$

Key Points to Remember

Essential concepts to master this topic

Recognition: Identify when a quadratic forms a perfect square trinomial
Completing the Square: Transform $x^2 + x + \frac{1}{4}$ into $(x + \frac{1}{2})^2 = 0$
Verification: Substitute $x = -\frac{1}{2}$ back: $\frac{1}{4} - \frac{1}{2} + \frac{1}{4} = 0$ ✓

Common Mistakes

Avoid these frequent errors

Using the quadratic formula unnecessarily
Don't automatically reach for the quadratic formula when $a = 1$ and the equation might be a perfect square! This leads to complicated calculations with discriminants. Always check if the equation can be factored as a perfect square first - it's much faster and cleaner.

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 $x^2 + 2bx + b^2$ . In this case, we have $x^2 + x + \frac{1}{4}$ , where b = 1/2 because $2 \cdot \frac{1}{2} = 1$ and $(\frac{1}{2})^2 = \frac{1}{4}$ .

What if the equation doesn't form a perfect square?

Then you'll need to use other methods like the quadratic formula or factoring. But always check for perfect squares first - they're the easiest to solve!

Why does this equation have only one solution?

When $(x + \frac{1}{2})^2 = 0$ , the only way a square can equal zero is if the expression inside equals zero. This gives us a repeated root or double root at $x = -\frac{1}{2}$ .

Can I solve this using the quadratic formula instead?

Yes, but it's more work! Using $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ with a=1, b=1, c=1/4 gives the same answer, but recognizing the perfect square is much faster.

How do I check my answer?

Substitute $x = -\frac{1}{2}$ into the original equation: $(-\frac{1}{2})^2 + (-\frac{1}{2}) + \frac{1}{4} = \frac{1}{4} - \frac{1}{2} + \frac{1}{4} = 0$ ✓

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