WordGPT

Solve -x² - 4x + 6 = 0

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

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 ≈ -5.16228, 1.16228

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 = -x² - 4x + 6-6-4-22-20-15-10-5510x = -5.1622777x = 1.1622777vertexy = -x² - 4x + 6
Graph of y = -x² - 4x + 6. The curve crosses the x-axis at x = -5.1622777 and x = 1.1622777; 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=(−4)2−4⋅(−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 ≈ -5.1622777 and x ≈ 1.1622777.

  7. 7

    Check numerically: for x ≈ -5.1622777 the left side is -3.5527137e-15 (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
-51
-46
-39
-210
-19
06
11

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 ≈ -5.1622777 and x ≈ 1.1622777.

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.

Similar equations

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