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Solve -x² + 4x + 6 = 0

The solutions of -x² + 4x + 6 = 0 are x = 2 + √10 or x = 2 - √10, that is x ≈ -1.1622777 and x ≈ 5.1622777.

x² +x += 0
−x2+4x+6=0⇒x=2+10, 2−10-x^2 + 4x + 6 = 0 \Rightarrow x = 2 + \sqrt{10},\ 2 - \sqrt{10}
Discriminant D = 40 · x ≈ -1.16228, 5.16228

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

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Graph of y = -x² + 4x + 6-2246-20-15-10-5510x = -1.1622777x = 5.1622777vertexy = -x² + 4x + 6
Graph of y = -x² + 4x + 6. The curve crosses the x-axis at x = -1.1622777 and x = 5.1622777; the vertex is at (2, 10).

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 = 4, c = 6.

    −x2+4x+6=0-x^2 + 4x + 6 = 0
  2. 2

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

    D=42−4⋅(−1)⋅6=16+24=40D = 4^2 - 4\cdot(-1)\cdot6 = 16 + 24 = 40
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−4±40−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{40}}{-2}
  5. 5

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

    x=2+10orx=2−10x = 2 + \sqrt{10}\quad\text{or}\quad x = 2 - \sqrt{10}
  6. 6

    As decimals: x ≈ -1.1622777 and x ≈ 5.1622777.

  7. 7

    Check numerically: for x ≈ -1.1622777 the left side is -8.8817842e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 4x + 6 = 0\ \Rightarrow\ x = 2 + \sqrt{10},\ 2 - \sqrt{10} \]
\[ -x^2 + 4x + 6 = 0 \]
\[ D = 4^2 - 4\cdot(-1)\cdot6 = 16 + 24 = 40 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{40}}{-2} \]
\[ x = 2 + \sqrt{10}\quad\text{or}\quad x = 2 - \sqrt{10} \]

The parabola y = -x² + 4x + 6

xy
-11
06
19
210
39
46
51

Questions

What are the solutions of -x² + 4x + 6 = 0?

The solutions of -x² + 4x + 6 = 0 are x = 2 + √10 or x = 2 - √10, that is x ≈ -1.1622777 and x ≈ 5.1622777.

Does -x² + 4x + 6 = 0 factor?

Not over the integers, because the discriminant 40 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 4x + 6?

(2, 10). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 4, product = c/a = -6.

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    Solve -x² + 4x + 6 = 0: x = 2 + √10 or x = 2 - √10 (steps) | WordGPT