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Solve 2x² - 12x + 7 = 0

The solutions of 2x² - 12x + 7 = 0 are x = (6 + √22)/2 or x = (6 - √22)/2, that is x ≈ 5.3452079 and x ≈ 0.65479212.

x² +x += 0
2x2−12x+7=0⇒x=6+222, 6−2222x^2 - 12x + 7 = 0 \Rightarrow x = \frac{6 + \sqrt{22}}{2},\ \frac{6 - \sqrt{22}}{2}
Discriminant D = 88 · x ≈ 5.34521, 0.654792

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

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Graph of y = 2x² - 12x + 7-11234567-10102030x = 5.3452079x = 0.65479212vertexy = 2x² - 12x + 7
Graph of y = 2x² - 12x + 7. The curve crosses the x-axis at x = 5.3452079 and x = 0.65479212; the vertex is at (3, -11).

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=88D = (-12)^2 - 4\cdot2\cdot7 = 144 - 56 = 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=12±884x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{88}}{4}
  5. 5

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

    x=6+222orx=6−222x = \frac{6 + \sqrt{22}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{22}}{2}
  6. 6

    As decimals: x ≈ 5.3452079 and x ≈ 0.65479212.

  7. 7

    Check numerically: for x ≈ 5.3452079 the left side is 7.1054274e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 12x + 7 = 0\ \Rightarrow\ x = \frac{6 + \sqrt{22}}{2},\ \frac{6 - \sqrt{22}}{2} \]
\[ 2x^2 - 12x + 7 = 0 \]
\[ D = (-12)^2 - 4\cdot2\cdot7 = 144 - 56 = 88 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{88}}{4} \]
\[ x = \frac{6 + \sqrt{22}}{2}\quad\text{or}\quad x = \frac{6 - \sqrt{22}}{2} \]

The parabola y = 2x² - 12x + 7

xy
07
1-3
2-9
3-11
4-9
5-3
67

Questions

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

The solutions of 2x² - 12x + 7 = 0 are x = (6 + √22)/2 or x = (6 - √22)/2, that is x ≈ 5.3452079 and x ≈ 0.65479212.

Does 2x² - 12x + 7 = 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 = 2x² - 12x + 7?

(3, -11). 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 + √22)/2 or x = (6 - √22)/2 (steps) | WordGPT