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Solve -x² + 6x - 3 = 0

The solutions of -x² + 6x - 3 = 0 are x = 3 + √6 or x = 3 - √6, that is x ≈ 0.55051026 and x ≈ 5.4494897.

x² +x += 0
−x2+6x−3=0⇒x=3+6, 3−6-x^2 + 6x - 3 = 0 \Rightarrow x = 3 + \sqrt{6},\ 3 - \sqrt{6}
Discriminant D = 24 · x ≈ 0.55051, 5.44949

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

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Graph of y = -x² + 6x - 3-11234567-20-15-10-55x = 0.55051026x = 5.4494897vertexy = -x² + 6x - 3
Graph of y = -x² + 6x - 3. The curve crosses the x-axis at x = 0.55051026 and x = 5.4494897; the vertex is at (3, 6).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = 6, c = -3.

    −x2+6x−3=0-x^2 + 6x - 3 = 0
  2. 2

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

    D=62−4⋅(−1)⋅(−3)=36−12=24D = 6^2 - 4\cdot(-1)\cdot(-3) = 36 - 12 = 24
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−6±24−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{24}}{-2}
  5. 5

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

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

    As decimals: x ≈ 0.55051026 and x ≈ 5.4494897.

  7. 7

    Check numerically: for x ≈ 0.55051026 the left side is 8.8817842e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 6x - 3 = 0\ \Rightarrow\ x = 3 + \sqrt{6},\ 3 - \sqrt{6} \]
\[ -x^2 + 6x - 3 = 0 \]
\[ D = 6^2 - 4\cdot(-1)\cdot(-3) = 36 - 12 = 24 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{24}}{-2} \]
\[ x = 3 + \sqrt{6}\quad\text{or}\quad x = 3 - \sqrt{6} \]

The parabola y = -x² + 6x - 3

xy
0-3
12
25
36
45
52
6-3

Questions

What are the solutions of -x² + 6x - 3 = 0?

The solutions of -x² + 6x - 3 = 0 are x = 3 + √6 or x = 3 - √6, that is x ≈ 0.55051026 and x ≈ 5.4494897.

Does -x² + 6x - 3 = 0 factor?

Not over the integers, because the discriminant 24 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 6x - 3?

(3, 6). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 6, product = c/a = 3.

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