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Solve 2x² - 12x - 7 = 0

The solutions of 2x² - 12x - 7 = 0 are x = (6 + 5√2)/2 or x = (6 - 5√2)/2, that is x ≈ 6.5355339 and x ≈ -0.53553391.

x² +x += 0
2x2−12x−7=0⇒x=6+522, 6−5222x^2 - 12x - 7 = 0 \Rightarrow x = \frac{6 + 5\sqrt{2}}{2},\ \frac{6 - 5\sqrt{2}}{2}
Discriminant D = 200 · x ≈ 6.53553, -0.535534

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

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Graph of y = 2x² - 12x - 7-22468-30-20-101020304050x = 6.5355339x = -0.53553391vertexy = 2x² - 12x - 7
Graph of y = 2x² - 12x - 7. The curve crosses the x-axis at x = 6.5355339 and x = -0.53553391; the vertex is at (3, -25).

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 = -7.

    2x2−12x−7=02x^2 - 12x - 7 = 0
  2. 2

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

    D=(−12)2−4⋅2⋅(−7)=144+56=200D = (-12)^2 - 4\cdot2\cdot(-7) = 144 + 56 = 200
  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±2004x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{200}}{4}
  5. 5

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

    x=6+522orx=6−522x = \frac{6 + 5\sqrt{2}}{2}\quad\text{or}\quad x = \frac{6 - 5\sqrt{2}}{2}
  6. 6

    As decimals: x ≈ 6.5355339 and x ≈ -0.53553391.

  7. 7

    Check numerically: for x ≈ 6.5355339 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 12x - 7 = 0\ \Rightarrow\ x = \frac{6 + 5\sqrt{2}}{2},\ \frac{6 - 5\sqrt{2}}{2} \]
\[ 2x^2 - 12x - 7 = 0 \]
\[ D = (-12)^2 - 4\cdot2\cdot(-7) = 144 + 56 = 200 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{200}}{4} \]
\[ x = \frac{6 + 5\sqrt{2}}{2}\quad\text{or}\quad x = \frac{6 - 5\sqrt{2}}{2} \]

The parabola y = 2x² - 12x - 7

xy
0-7
1-17
2-23
3-25
4-23
5-17
6-7

Questions

What are the solutions of 2x² - 12x - 7 = 0?

The solutions of 2x² - 12x - 7 = 0 are x = (6 + 5√2)/2 or x = (6 - 5√2)/2, that is x ≈ 6.5355339 and x ≈ -0.53553391.

Does 2x² - 12x - 7 = 0 factor?

Not over the integers, because the discriminant 200 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 12x - 7?

(3, -25). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 6, product = c/a = -7/2.

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    Solve 2x² - 12x - 7 = 0: x = (6 + 5√2)/2 or x = (6 - 5√2)/2 (steps) | WordGPT