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Solve x² + 4x + 4 = 0

The solution of x² + 4x + 4 = 0 is x = -2 (a repeated root).

x² +x += 0
x2+4x+4=0⇒x=−2x^2 + 4x + 4 = 0 \Rightarrow x = -2
Discriminant D = 0 · x ≈ -2

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

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Graph of y = x² + 4x + 4-4.5-4-3.5-3-2.5-2-1.5-1-0.50.5123456x = -2vertexy = x² + 4x + 4
Graph of y = x² + 4x + 4. The curve crosses the x-axis at x = -2; the vertex is at (-2, 0).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 1, b = 4, c = 4.

    x2+4x+4=0x^2 + 4x + 4 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=42−4⋅1⋅4=16−16=0D = 4^2 - 4\cdot1\cdot4 = 16 - 16 = 0
  3. 3

    D is zero, so there is exactly one solution (a repeated root).

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    (x+2)(x+2)=0\left(x + 2\right)\left(x + 2\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x+2=0⇒x=−2x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−4±02x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{0}}{2}
  7. 7

    Simplify to get the solution:

    x=−2x = -2
  8. 8

    Check by putting x = -2 back in: a·x² + b·x + c = 1·(4) + 4·(-2) + 4 = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 + 4x + 4 = 0\ \Rightarrow\ x = -2 \]
\[ x^2 + 4x + 4 = 0 \]
\[ D = 4^2 - 4\cdot1\cdot4 = 16 - 16 = 0 \]
\[ \left(x + 2\right)\left(x + 2\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{0}}{2} \]
\[ x = -2 \]

The parabola y = x² + 4x + 4

xy
-59
-44
-31
-20
-11
04
19

Questions

What are the solutions of x² + 4x + 4 = 0?

The solution of x² + 4x + 4 = 0 is x = -2 (a repeated root).

Does x² + 4x + 4 = 0 factor?

Yes: (x+2)(x+2) = 0.

What is the vertex of y = x² + 4x + 4?

(-2, 0). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -4, product = c/a = 4.

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