WordGPT

Solve -3x² + 1 = 0

The solutions of -3x² + 1 = 0 are x = (√3)/3 or x = (-√3)/3, that is x ≈ -0.57735027 and x ≈ 0.57735027.

x² +x += 0
−3x2+1=0⇒x=33, −33-3x^2 + 1 = 0 \Rightarrow x = \frac{\sqrt{3}}{3},\ \frac{-\sqrt{3}}{3}
Discriminant D = 12 · x ≈ -0.57735, 0.57735

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 + 1-3-2-1123-25-20-15-10-5x = -0.57735027x = 0.57735027vertexy = -3x² + 1
Graph of y = -3x² + 1. The curve crosses the x-axis at x = -0.57735027 and x = 0.57735027; the vertex is at (0, 1).

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 = 0, c = 1.

    −3x2+1=0-3x^2 + 1 = 0
  2. 2

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

    D=02−4⋅(−3)⋅1=0+12=12D = 0^2 - 4\cdot(-3)\cdot1 = 0 + 12 = 12
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=0±12−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{0 \pm \sqrt{12}}{-6}
  5. 5

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

    x=33orx=−33x = \frac{\sqrt{3}}{3}\quad\text{or}\quad x = \frac{-\sqrt{3}}{3}
  6. 6

    As decimals: x ≈ -0.57735027 and x ≈ 0.57735027.

  7. 7

    Check numerically: for x ≈ -0.57735027 the left side is 1.110223e-16 (zero up to rounding) ✓

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

The parabola y = -3x² + 1

xy
-3-26
-2-11
-1-2
01
1-2
2-11
3-26

Questions

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

The solutions of -3x² + 1 = 0 are x = (√3)/3 or x = (-√3)/3, that is x ≈ -0.57735027 and x ≈ 0.57735027.

Does -3x² + 1 = 0 factor?

Not over the integers, because the discriminant 12 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -3x² + 1?

(0, 1). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 0, product = c/a = -1/3.

Similar equations

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