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Solve -3x² + 9x - 3 = 0

The solutions of -3x² + 9x - 3 = 0 are x = (3 + √5)/2 or x = (3 - √5)/2, that is x ≈ 0.38196601 and x ≈ 2.618034.

x² +x += 0
−3x2+9x−3=0⇒x=3+52, 3−52-3x^2 + 9x - 3 = 0 \Rightarrow x = \frac{3 + \sqrt{5}}{2},\ \frac{3 - \sqrt{5}}{2}
Discriminant D = 45 · x ≈ 0.381966, 2.61803

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

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Graph of y = -3x² + 9x - 3-2-112345-30-20-10x = 0.38196601x = 2.618034vertexy = -3x² + 9x - 3
Graph of y = -3x² + 9x - 3. The curve crosses the x-axis at x = 0.38196601 and x = 2.618034; the vertex is at (1.5, 3.75).

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 = 9, c = -3.

    −3x2+9x−3=0-3x^2 + 9x - 3 = 0
  2. 2

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

    D=92−4⋅(−3)⋅(−3)=81−36=45D = 9^2 - 4\cdot(-3)\cdot(-3) = 81 - 36 = 45
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−9±45−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{45}}{-6}
  5. 5

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

    x=3+52orx=3−52x = \frac{3 + \sqrt{5}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{5}}{2}
  6. 6

    As decimals: x ≈ 0.38196601 and x ≈ 2.618034.

  7. 7

    Check numerically: for x ≈ 0.38196601 the left side is -4.4408921e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 9x - 3 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{5}}{2},\ \frac{3 - \sqrt{5}}{2} \]
\[ -3x^2 + 9x - 3 = 0 \]
\[ D = 9^2 - 4\cdot(-3)\cdot(-3) = 81 - 36 = 45 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{45}}{-6} \]
\[ x = \frac{3 + \sqrt{5}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{5}}{2} \]

The parabola y = -3x² + 9x - 3

xy
-1.5-23.25
-0.5-8.25
0.50.75
1.53.75
2.50.75
3.5-8.25
4.5-23.25

Questions

What are the solutions of -3x² + 9x - 3 = 0?

The solutions of -3x² + 9x - 3 = 0 are x = (3 + √5)/2 or x = (3 - √5)/2, that is x ≈ 0.38196601 and x ≈ 2.618034.

Does -3x² + 9x - 3 = 0 factor?

Not over the integers, because the discriminant 45 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -3x² + 9x - 3?

(3/2, 15/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = 1.

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