Solve the Quadratic Equation: 4x² - 4x + 1 = 0

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

When $ b^2 - 4ac = 0 $, there is exactly one repeated root. This means the parabola just touches the x-axis at one point instead of crossing it twice.

Q: Can I factor this equation instead of using the quadratic formula?

Yes! This equation is actually a perfect square trinomial: $ 4x^2 - 4x + 1 = (2x - 1)^2 = 0 $. Setting $ 2x - 1 = 0 $ gives $ x = \frac{1}{2} $.

Q: Why do I get the same answer twice when using the quadratic formula?

When the discriminant is zero, $ \pm \sqrt{0} = \pm 0 = 0 $. So both plus and minus versions give the same result - that's your repeated root!

Q: How can I recognize a perfect square trinomial?

Look for the pattern $ a^2 - 2ab + b^2 $. Here: $ (2x)^2 - 2(2x)(1) + 1^2 = 4x^2 - 4x + 1 $. The middle term is always twice the product of the square roots!

Q: What if I made an arithmetic error in the discriminant?

Double-check each step: $ (-4)^2 = 16 $, then $ 4 \times 4 \times 1 = 16 $, so $ 16 - 16 = 0 $. A common mistake is forgetting that negative times negative equals positive!

Quadratic Equations with Zero Discriminant

Solve the following equation:

$4x^2-4x+1=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find the value of X.

00:09 We'll start by using the roots formula.

00:34 First, identify the coefficients from the equation.

00:50 Now, substitute the given values and begin solving.

01:14 Calculate the square and the products carefully.

01:34 Remember, the square root of zero is always zero.

01:39 When the root equals zero, it means there's only one solution.

02:01 And that's how we find the answer to this question.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$4x^2-4x+1=0$

Step-by-step solution

To solve the equation $4x^2 - 4x + 1 = 0$ , we will use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

First, we identify $a = 4$ , $b = -4$ , and $c = 1$ .

Calculate the discriminant:

$b^2 - 4ac = (-4)^2 - 4 \times 4 \times 1 = 16 - 16 = 0$

Since the discriminant is 0, there is one real repeated root.

Substitute into the quadratic formula:

$x = \frac{-(-4) \pm \sqrt{0}}{2 \times 4} = \frac{4}{8} = \frac{1}{2}$

Therefore, the solution to the equation is $x = \frac{1}{2}$ .

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Formula: Use quadratic formula when factoring is difficult
Discriminant: Calculate $b^2 - 4ac = (-4)^2 - 4(4)(1) = 0$
Check: Substitute $x = \frac{1}{2}$ : $4(\frac{1}{2})^2 - 4(\frac{1}{2}) + 1 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative sign when substituting b = -4
Don't substitute -(-4) as just 4 = wrong calculation! This gives you -4 in the numerator instead of +4, leading to negative solutions. Always be extra careful with negative coefficients and use parentheses: -(-4) = +4.

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 equals zero?

When $b^2 - 4ac = 0$ , there is exactly one repeated root. This means the parabola just touches the x-axis at one point instead of crossing it twice.

Can I factor this equation instead of using the quadratic formula?

Yes! This equation is actually a perfect square trinomial: $4x^2 - 4x + 1 = (2x - 1)^2 = 0$ . Setting $2x - 1 = 0$ gives $x = \frac{1}{2}$ .

Why do I get the same answer twice when using the quadratic formula?

When the discriminant is zero, $\pm \sqrt{0} = \pm 0 = 0$ . So both plus and minus versions give the same result - that's your repeated root!

How can I recognize a perfect square trinomial?

Look for the pattern $a^2 - 2ab + b^2$ . Here: $(2x)^2 - 2(2x)(1) + 1^2 = 4x^2 - 4x + 1$ . The middle term is always twice the product of the square roots!

What if I made an arithmetic error in the discriminant?

Double-check each step: $(-4)^2 = 16$ , then $4 \times 4 \times 1 = 16$ , so $16 - 16 = 0$ . A common mistake is forgetting that negative times negative equals positive!

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