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Solve 3x² + 6x + 3 = 0

The solution of 3x² + 6x + 3 = 0 is x = -1 (a repeated root).

x² +x += 0
3x2+6x+3=0⇒x=−13x^2 + 6x + 3 = 0 \Rightarrow x = -1
Discriminant D = 0 · x ≈ -1

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

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Graph of y = 3x² + 6x + 3-3.5-3-2.5-2-1.5-1-0.50.511.55101520x = -1vertexy = 3x² + 6x + 3
Graph of y = 3x² + 6x + 3. The curve crosses the x-axis at x = -1; the vertex is at (-1, 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 = 3, b = 6, c = 3.

    3x2+6x+3=03x^2 + 6x + 3 = 0
  2. 2

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

    D=62−4⋅3⋅3=36−36=0D = 6^2 - 4\cdot3\cdot3 = 36 - 36 = 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:

    3(x+1)(x+1)=03\left(x + 1\right)\left(x + 1\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+1=0⇒x=−1x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−6±06x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{0}}{6}
  7. 7

    Simplify to get the solution:

    x=−1x = -1
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = 3·(1) + 6·(-1) + 3 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 6x + 3 = 0\ \Rightarrow\ x = -1 \]
\[ 3x^2 + 6x + 3 = 0 \]
\[ D = 6^2 - 4\cdot3\cdot3 = 36 - 36 = 0 \]
\[ 3\left(x + 1\right)\left(x + 1\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{0}}{6} \]
\[ x = -1 \]

The parabola y = 3x² + 6x + 3

xy
-427
-312
-23
-10
03
112
227

Questions

What are the solutions of 3x² + 6x + 3 = 0?

The solution of 3x² + 6x + 3 = 0 is x = -1 (a repeated root).

Does 3x² + 6x + 3 = 0 factor?

Yes: 3(x+1)(x+1) = 0.

What is the vertex of y = 3x² + 6x + 3?

(-1, 0). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -2, product = c/a = 1.

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