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Solve -x² - 6x + 3 = 0

The solutions of -x² - 6x + 3 = 0 are x = -3 + 2√3 or x = -3 - 2√3, that is x ≈ -6.4641016 and x ≈ 0.46410162.

x² +x += 0
−x2−6x+3=0⇒x=−3+23, −3−23-x^2 - 6x + 3 = 0 \Rightarrow x = -3 + 2\sqrt{3},\ -3 - 2\sqrt{3}
Discriminant D = 48 · x ≈ -6.4641, 0.464102

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

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Graph of y = -x² - 6x + 3-8-6-4-22-25-20-15-10-551015x = -6.4641016x = 0.46410162vertexy = -x² - 6x + 3
Graph of y = -x² - 6x + 3. The curve crosses the x-axis at x = -6.4641016 and x = 0.46410162; the vertex is at (-3, 12).

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=(−6)2−4⋅(−1)⋅3=36+12=48D = (-6)^2 - 4\cdot(-1)\cdot3 = 36 + 12 = 48
  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±48−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{48}}{-2}
  5. 5

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

    x=−3+23orx=−3−23x = -3 + 2\sqrt{3}\quad\text{or}\quad x = -3 - 2\sqrt{3}
  6. 6

    As decimals: x ≈ -6.4641016 and x ≈ 0.46410162.

  7. 7

    Check numerically: for x ≈ -6.4641016 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 - 6x + 3 = 0\ \Rightarrow\ x = -3 + 2\sqrt{3},\ -3 - 2\sqrt{3} \]
\[ -x^2 - 6x + 3 = 0 \]
\[ D = (-6)^2 - 4\cdot(-1)\cdot3 = 36 + 12 = 48 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{48}}{-2} \]
\[ x = -3 + 2\sqrt{3}\quad\text{or}\quad x = -3 - 2\sqrt{3} \]

The parabola y = -x² - 6x + 3

xy
-63
-58
-411
-312
-211
-18
03

Questions

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

The solutions of -x² - 6x + 3 = 0 are x = -3 + 2√3 or x = -3 - 2√3, that is x ≈ -6.4641016 and x ≈ 0.46410162.

Does -x² - 6x + 3 = 0 factor?

Not over the integers, because the discriminant 48 is not a perfect square. Use the quadratic formula.

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

(-3, 12). 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 + 2√3 or x = -3 - 2√3 (steps) | WordGPT