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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=−1-3x^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 - 3-3.5-3-2.5-2-1.5-1-0.50.511.5-20-15-10-5x = -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=0-3x^2 - 6x - 3 = 0
  2. 2

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

    D=(−6)2−4⋅(−3)⋅(−3)=36−36=0D = (-6)^2 - 4\cdot(-3)\cdot(-3) = 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)=0-3\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±0−6x = \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
\[ -3x^2 - 6x - 3 = 0\ \Rightarrow\ x = -1 \]
\[ -3x^2 - 6x - 3 = 0 \]
\[ D = (-6)^2 - 4\cdot(-3)\cdot(-3) = 36 - 36 = 0 \]
\[ -3\left(x + 1\right)\left(x + 1\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{0}}{-6} \]
\[ x = -1 \]

The parabola y = -3x² - 6x - 3

xy
-4-27
-3-12
-2-3
-10
0-3
1-12
2-27

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 downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -2, product = c/a = 1.

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