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Solve -3x² - 2x + 1 = 0

The solutions of -3x² - 2x + 1 = 0 are x = -1 or x = 1/3.

x² +x += 0
−3x2−2x+1=0⇒x=−1, 13-3x^2 - 2x + 1 = 0 \Rightarrow x = -1,\ \frac{1}{3}
Discriminant D = 16 · x ≈ -1, 0.333333

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

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Graph of y = -3x² - 2x + 1-3-2-112-30-25-20-15-10-55x = -1x = 0.33333333vertexy = -3x² - 2x + 1
Graph of y = -3x² - 2x + 1. The curve crosses the x-axis at x = -1 and x = 0.33333333; the vertex is at (-0.33333333, 1.3333333).

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 = -2, c = 1.

    −3x2−2x+1=0-3x^2 - 2x + 1 = 0
  2. 2

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

    D=(−2)2−4⋅(−3)⋅1=4+12=16D = (-2)^2 - 4\cdot(-3)\cdot1 = 4 + 12 = 16
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

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

    −(x+1)(3x−1)=0-\left(x + 1\right)\left(3x - 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=−13x−1=0⇒x=13x + 1 = 0 \Rightarrow x = -1\qquad 3x - 1 = 0 \Rightarrow x = \frac{1}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=2±16−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{16}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−1orx=13x = -1\quad\text{or}\quad x = \frac{1}{3}
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = -3·(1) + (-2)·(-1) + 1 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 2x + 1 = 0\ \Rightarrow\ x = -1,\ \frac{1}{3} \]
\[ -3x^2 - 2x + 1 = 0 \]
\[ D = (-2)^2 - 4\cdot(-3)\cdot1 = 4 + 12 = 16 \]
\[ -\left(x + 1\right)\left(3x - 1\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 3x - 1 = 0 \Rightarrow x = \frac{1}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{16}}{-6} \]
\[ x = -1\quad\text{or}\quad x = \frac{1}{3} \]

The parabola y = -3x² - 2x + 1

xy
-3.3333-25.666067
-2.3333-10.666267
-1.3333-1.6664667
-0.33331.3333333
0.6667-1.6668667
1.6667-10.667067
2.6667-25.667267

Questions

What are the solutions of -3x² - 2x + 1 = 0?

The solutions of -3x² - 2x + 1 = 0 are x = -1 or x = 1/3.

Does -3x² - 2x + 1 = 0 factor?

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

What is the vertex of y = -3x² - 2x + 1?

(-1/3, 4/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -2/3, product = c/a = -1/3.

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