WordGPT

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 ≈ -1.2637626 and x ≈ 0.26376262.

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 ≈ -1.26376, 0.263763

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

Explain it step by step in chatMake a short video of it
Graph of y = -3x² - 3x + 1-3-2-112-30-25-20-15-10-55x = -1.2637626x = 0.26376262vertexy = -3x² - 3x + 1
Graph of y = -3x² - 3x + 1. The curve crosses the x-axis at x = -1.2637626 and x = 0.26376262; 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=(−3)2−4⋅(−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 ≈ -1.2637626 and x ≈ 0.26376262.

  7. 7

    Check numerically: for x ≈ -1.2637626 the left side is 0 (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
-3.5-25.25
-2.5-10.25
-1.5-1.25
-0.51.75
0.5-1.25
1.5-10.25
2.5-25.25

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 ≈ -1.2637626 and x ≈ 0.26376262.

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.

Similar equations

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