WordGPT

Solve -x² + 8x + 6 = 0

The solutions of -x² + 8x + 6 = 0 are x = 4 + √22 or x = 4 - √22, that is x ≈ -0.69041576 and x ≈ 8.6904158.

x² +x += 0
−x2+8x+6=0⇒x=4+22, 4−22-x^2 + 8x + 6 = 0 \Rightarrow x = 4 + \sqrt{22},\ 4 - \sqrt{22}
Discriminant D = 88 · x ≈ -0.690416, 8.69042

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 + 8x + 6-2246810-30-20-101020x = -0.69041576x = 8.6904158vertexy = -x² + 8x + 6
Graph of y = -x² + 8x + 6. The curve crosses the x-axis at x = -0.69041576 and x = 8.6904158; the vertex is at (4, 22).

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 = 8, c = 6.

    −x2+8x+6=0-x^2 + 8x + 6 = 0
  2. 2

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

    D=82−4⋅(−1)⋅6=64+24=88D = 8^2 - 4\cdot(-1)\cdot6 = 64 + 24 = 88
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−8±88−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{88}}{-2}
  5. 5

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

    x=4+22orx=4−22x = 4 + \sqrt{22}\quad\text{or}\quad x = 4 - \sqrt{22}
  6. 6

    As decimals: x ≈ -0.69041576 and x ≈ 8.6904158.

  7. 7

    Check numerically: for x ≈ -0.69041576 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 8x + 6 = 0\ \Rightarrow\ x = 4 + \sqrt{22},\ 4 - \sqrt{22} \]
\[ -x^2 + 8x + 6 = 0 \]
\[ D = 8^2 - 4\cdot(-1)\cdot6 = 64 + 24 = 88 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{88}}{-2} \]
\[ x = 4 + \sqrt{22}\quad\text{or}\quad x = 4 - \sqrt{22} \]

The parabola y = -x² + 8x + 6

xy
113
218
321
422
521
618
713

Questions

What are the solutions of -x² + 8x + 6 = 0?

The solutions of -x² + 8x + 6 = 0 are x = 4 + √22 or x = 4 - √22, that is x ≈ -0.69041576 and x ≈ 8.6904158.

Does -x² + 8x + 6 = 0 factor?

Not over the integers, because the discriminant 88 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 8x + 6?

(4, 22). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 8, product = c/a = -6.

Similar equations

    Solve -x² + 8x + 6 = 0: x = 4 + √22 or x = 4 - √22 (steps) | WordGPT