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Solve -3x² - 4x + 1 = 0

The solutions of -3x² - 4x + 1 = 0 are x = (-2 + √7)/3 or x = (-2 - √7)/3, that is x ≈ -1.5485838 and x ≈ 0.21525044.

x² +x += 0
−3x2−4x+1=0⇒x=−2+73, −2−73-3x^2 - 4x + 1 = 0 \Rightarrow x = \frac{-2 + \sqrt{7}}{3},\ \frac{-2 - \sqrt{7}}{3}
Discriminant D = 28 · x ≈ -1.54858, 0.21525

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

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Graph of y = -3x² - 4x + 1-4-3-2-112-30-25-20-15-10-55x = -1.5485838x = 0.21525044vertexy = -3x² - 4x + 1
Graph of y = -3x² - 4x + 1. The curve crosses the x-axis at x = -1.5485838 and x = 0.21525044; the vertex is at (-0.66666667, 2.3333333).

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 = -4, c = 1.

    −3x2−4x+1=0-3x^2 - 4x + 1 = 0
  2. 2

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

    D=(−4)2−4⋅(−3)⋅1=16+12=28D = (-4)^2 - 4\cdot(-3)\cdot1 = 16 + 12 = 28
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=4±28−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{28}}{-6}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=−2+73orx=−2−73x = \frac{-2 + \sqrt{7}}{3}\quad\text{or}\quad x = \frac{-2 - \sqrt{7}}{3}
  6. 6

    As decimals: x ≈ -1.5485838 and x ≈ 0.21525044.

  7. 7

    Check numerically: for x ≈ -1.5485838 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 4x + 1 = 0\ \Rightarrow\ x = \frac{-2 + \sqrt{7}}{3},\ \frac{-2 - \sqrt{7}}{3} \]
\[ -3x^2 - 4x + 1 = 0 \]
\[ D = (-4)^2 - 4\cdot(-3)\cdot1 = 16 + 12 = 28 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{28}}{-6} \]
\[ x = \frac{-2 + \sqrt{7}}{3}\quad\text{or}\quad x = \frac{-2 - \sqrt{7}}{3} \]

The parabola y = -3x² - 4x + 1

xy
-3.6667-24.667267
-2.6667-9.6670667
-1.6667-0.66686667
-0.66672.3333333
0.3333-0.66646667
1.3333-9.6662667
2.3333-24.666067

Questions

What are the solutions of -3x² - 4x + 1 = 0?

The solutions of -3x² - 4x + 1 = 0 are x = (-2 + √7)/3 or x = (-2 - √7)/3, that is x ≈ -1.5485838 and x ≈ 0.21525044.

Does -3x² - 4x + 1 = 0 factor?

Not over the integers, because the discriminant 28 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -3x² - 4x + 1?

(-2/3, 7/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -4/3, product = c/a = -1/3.

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    Solve -3x² - 4x + 1 = 0: x = (-2 + √7)/3 or x = (-2 - √7)/3 (steps) | WordGPT