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Solve 2x² - 3x - 11 = 0

The solutions of 2x² - 3x - 11 = 0 are x = (3 + √97)/4 or x = (3 - √97)/4, that is x ≈ 3.2122145 and x ≈ -1.7122145.

x² +x += 0
2x2−3x−11=0⇒x=3+974, 3−9742x^2 - 3x - 11 = 0 \Rightarrow x = \frac{3 + \sqrt{97}}{4},\ \frac{3 - \sqrt{97}}{4}
Discriminant D = 97 · x ≈ 3.21221, -1.71221

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

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Graph of y = 2x² - 3x - 11-4-3-2-112345-1010203040x = 3.2122145x = -1.7122145vertexy = 2x² - 3x - 11
Graph of y = 2x² - 3x - 11. The curve crosses the x-axis at x = 3.2122145 and x = -1.7122145; the vertex is at (0.75, -12.125).

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 = -3, c = -11.

    2x2−3x−11=02x^2 - 3x - 11 = 0
  2. 2

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

    D=(−3)2−4⋅2⋅(−11)=9+88=97D = (-3)^2 - 4\cdot2\cdot(-11) = 9 + 88 = 97
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=3±974x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{97}}{4}
  5. 5

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

    x=3+974orx=3−974x = \frac{3 + \sqrt{97}}{4}\quad\text{or}\quad x = \frac{3 - \sqrt{97}}{4}
  6. 6

    As decimals: x ≈ 3.2122145 and x ≈ -1.7122145.

  7. 7

    Check numerically: for x ≈ 3.2122145 the left side is -3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 3x - 11 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{97}}{4},\ \frac{3 - \sqrt{97}}{4} \]
\[ 2x^2 - 3x - 11 = 0 \]
\[ D = (-3)^2 - 4\cdot2\cdot(-11) = 9 + 88 = 97 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{97}}{4} \]
\[ x = \frac{3 + \sqrt{97}}{4}\quad\text{or}\quad x = \frac{3 - \sqrt{97}}{4} \]

The parabola y = 2x² - 3x - 11

xy
-2.255.875
-1.25-4.125
-0.25-10.125
0.75-12.125
1.75-10.125
2.75-4.125
3.755.875

Questions

What are the solutions of 2x² - 3x - 11 = 0?

The solutions of 2x² - 3x - 11 = 0 are x = (3 + √97)/4 or x = (3 - √97)/4, that is x ≈ 3.2122145 and x ≈ -1.7122145.

Does 2x² - 3x - 11 = 0 factor?

Not over the integers, because the discriminant 97 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 3x - 11?

(3/4, -97/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 3/2, product = c/a = -11/2.

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    Solve 2x² - 3x - 11 = 0: x = (3 + √97)/4 or x = (3 - √97)/4 (steps) | WordGPT