WordGPT

Solve -2x² + 12x + 1 = 0

The solutions of -2x² + 12x + 1 = 0 are x = (6 + √38)/2 or x = (6 - √38)/2, that is x ≈ -0.082207001 and x ≈ 6.082207.

x² +x += 0
−2x2+12x+1=0⇒x=6+382, 6−382-2x^2 + 12x + 1 = 0 \Rightarrow x = \frac{6 + \sqrt{38}}{2},\ \frac{6 - \sqrt{38}}{2}
Discriminant D = 152 · x ≈ -0.082207, 6.08221

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 = -2x² + 12x + 1-22468-40-30-20-101020x = -0.082207001x = 6.082207vertexy = -2x² + 12x + 1
Graph of y = -2x² + 12x + 1. The curve crosses the x-axis at x = -0.082207001 and x = 6.082207; the vertex is at (3, 19).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -2, b = 12, c = 1.

    −2x2+12x+1=0-2x^2 + 12x + 1 = 0
  2. 2

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

    D=122−4⋅(−2)⋅1=144+8=152D = 12^2 - 4\cdot(-2)\cdot1 = 144 + 8 = 152
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−12±152−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{152}}{-4}
  5. 5

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

    x=6+382orx=6−382x = \frac{6 + \sqrt{38}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{38}}{2}
  6. 6

    As decimals: x ≈ -0.082207001 and x ≈ 6.082207.

  7. 7

    Check numerically: for x ≈ -0.082207001 the left side is 2.6645353e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 12x + 1 = 0\ \Rightarrow\ x = \frac{6 + \sqrt{38}}{2},\ \frac{6 - \sqrt{38}}{2} \]
\[ -2x^2 + 12x + 1 = 0 \]
\[ D = 12^2 - 4\cdot(-2)\cdot1 = 144 + 8 = 152 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{152}}{-4} \]
\[ x = \frac{6 + \sqrt{38}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{38}}{2} \]

The parabola y = -2x² + 12x + 1

xy
01
111
217
319
417
511
61

Questions

What are the solutions of -2x² + 12x + 1 = 0?

The solutions of -2x² + 12x + 1 = 0 are x = (6 + √38)/2 or x = (6 - √38)/2, that is x ≈ -0.082207001 and x ≈ 6.082207.

Does -2x² + 12x + 1 = 0 factor?

Not over the integers, because the discriminant 152 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² + 12x + 1?

(3, 19). 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 = -1/2.

Similar equations

    Solve -2x² + 12x + 1 = 0: x = (6 + √38)/2 or x = (6 - √38)/2 (steps) | WordGPT