Solve the Quadratic Equation: X²+4X+4=0 Step-by-Step

Q: Why does this quadratic have only one solution instead of two?

This equation actually has one repeated solution (called a double root). When $ (X+2)^2 = 0 $, both factors are the same, so $ X = -2 $ appears twice.

Q: How do I recognize a perfect square trinomial?

Look for the pattern $ a^2 + 2ab + b^2 $. In $ X^2 + 4X + 4 $: first term is $ X^2 $, last term is $ 2^2 = 4 $, and middle term is $ 2(X)(2) = 4X $.

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

Yes! Using $ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $ gives $ x = \frac{-4 \pm \sqrt{16-16}}{2} = \frac{-4 \pm 0}{2} = -2 $. Notice the discriminant is zero, confirming one repeated solution.

Q: What does it mean graphically when there's one solution?

The parabola $ y = X^2 + 4X + 4 $ touches the x-axis at exactly one point $ (-2, 0) $. This point is called the vertex of the parabola.

Q: How do I check my answer is correct?

Substitute $ X = -2 $ into the original equation: $ (-2)^2 + 4(-2) + 4 = 4 - 8 + 4 = 0 $ ✓

Perfect Square Trinomials with Repeated Solutions

Solve for X:

$X^2+4X+4=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Note the coefficients, each number is as if multiplied by 1

00:10 Use the roots formula

00:19 Substitute appropriate values according to the given data and solve for X

00:25 Calculate the products and the square

00:35 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 for X:

$X^2+4X+4=0$

Step-by-step solution

To solve the quadratic equation $X^2 + 4X + 4 = 0$ , we can attempt to factor it as a perfect square trinomial.

Step 1: Recognize that the equation is a perfect square trinomial because it can be written as $(X + 2)^2$ . The relationship is $(X + 2)(X + 2) = X^2 + 4X + 4$ .
Step 2: Set the factored equation equal to zero: $(X + 2)^2 = 0$ .
Step 3: Solve for $X$ by setting $X + 2 = 0$ .
Step 4: Solving $X + 2 = 0$ gives $X = -2$ .

Therefore, the solution to the equation is $X = -2$ .

Thus, the correct answer choice is $x = -2$ , corresponding to choice 1.

Final Answer

$x=-2$

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Look for $a^2 + 2ab + b^2 = (a+b)^2$ form
Factoring: $X^2 + 4X + 4 = (X+2)^2$ because middle term is 2(X)(2)
Verification: Expand $(X+2)^2 = X^2 + 4X + 4$ to confirm ✓

Common Mistakes

Avoid these frequent errors

Expecting two different solutions from perfect square trinomials
Don't assume quadratic equations always have two different roots like $x_1 = -2, x_2 = 4$ ! Perfect squares give repeated roots because $(X+2)^2 = 0$ means $X+2 = 0$ twice. Always recognize that $X = -2$ is a double root.

Practice Quiz

Test your knowledge with interactive questions

What is the value of X in the following equation?

$$ X^2+10X+9=0 $$

FAQ

Everything you need to know about this question

Why does this quadratic have only one solution instead of two?

This equation actually has one repeated solution (called a double root). When $(X+2)^2 = 0$ , both factors are the same, so $X = -2$ appears twice.

How do I recognize a perfect square trinomial?

Look for the pattern $a^2 + 2ab + b^2$ . In $X^2 + 4X + 4$ : first term is $X^2$ , last term is $2^2 = 4$ , and middle term is $2(X)(2) = 4X$ .

Can I solve this using the quadratic formula instead?

Yes! Using $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ gives $x = \frac{-4 \pm \sqrt{16-16}}{2} = \frac{-4 \pm 0}{2} = -2$ . Notice the discriminant is zero, confirming one repeated solution.

What does it mean graphically when there's one solution?

The parabola $y = X^2 + 4X + 4$ touches the x-axis at exactly one point $(-2, 0)$ . This point is called the vertex of the parabola.

How do I check my answer is correct?

Substitute $X = -2$ into the original equation: $(-2)^2 + 4(-2) + 4 = 4 - 8 + 4 = 0$ ✓

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