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Solve -3x² + 3x + 1 = 0

The solutions of -3x² + 3x + 1 = 0 are x = (3 + √21)/6 or x = (3 - √21)/6, that is x ≈ -0.26376262 and x ≈ 1.2637626.

x² +x += 0
−3x2+3x+1=0⇒x=3+216, 3−216-3x^2 + 3x + 1 = 0 \Rightarrow x = \frac{3 + \sqrt{21}}{6},\ \frac{3 - \sqrt{21}}{6}
Discriminant D = 21 · x ≈ -0.263763, 1.26376

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

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Graph of y = -3x² + 3x + 1-2-1123-30-25-20-15-10-55x = -0.26376262x = 1.2637626vertexy = -3x² + 3x + 1
Graph of y = -3x² + 3x + 1. The curve crosses the x-axis at x = -0.26376262 and x = 1.2637626; the vertex is at (0.5, 1.75).

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 = 3, c = 1.

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

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

    D=32−4⋅(−3)⋅1=9+12=21D = 3^2 - 4\cdot(-3)\cdot1 = 9 + 12 = 21
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−3±21−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{21}}{-6}
  5. 5

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

    x=3+216orx=3−216x = \frac{3 + \sqrt{21}}{6}\quad\text{or}\quad x = \frac{3 - \sqrt{21}}{6}
  6. 6

    As decimals: x ≈ -0.26376262 and x ≈ 1.2637626.

  7. 7

    Check numerically: for x ≈ -0.26376262 the left side is 1.110223e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 3x + 1 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{21}}{6},\ \frac{3 - \sqrt{21}}{6} \]
\[ -3x^2 + 3x + 1 = 0 \]
\[ D = 3^2 - 4\cdot(-3)\cdot1 = 9 + 12 = 21 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{21}}{-6} \]
\[ x = \frac{3 + \sqrt{21}}{6}\quad\text{or}\quad x = \frac{3 - \sqrt{21}}{6} \]

The parabola y = -3x² + 3x + 1

xy
-2.5-25.25
-1.5-10.25
-0.5-1.25
0.51.75
1.5-1.25
2.5-10.25
3.5-25.25

Questions

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

The solutions of -3x² + 3x + 1 = 0 are x = (3 + √21)/6 or x = (3 - √21)/6, that is x ≈ -0.26376262 and x ≈ 1.2637626.

Does -3x² + 3x + 1 = 0 factor?

Not over the integers, because the discriminant 21 is not a perfect square. Use the quadratic formula.

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

(1/2, 7/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = -1/3.

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    Solve -3x² + 3x + 1 = 0: x = (3 + √21)/6 or x = (3 - √21)/6 (steps) | WordGPT