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Solve 2x² - 11x - 9 = 0

The solutions of 2x² - 11x - 9 = 0 are x = (11 + √193)/4 or x = (11 - √193)/4, that is x ≈ 6.223111 and x ≈ -0.723111.

x² +x += 0
2x2−11x−9=0⇒x=11+1934, 11−19342x^2 - 11x - 9 = 0 \Rightarrow x = \frac{11 + \sqrt{193}}{4},\ \frac{11 - \sqrt{193}}{4}
Discriminant D = 193 · x ≈ 6.22311, -0.723111

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

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Graph of y = 2x² - 11x - 9-22468-30-20-101020304050x = 6.223111x = -0.723111vertexy = 2x² - 11x - 9
Graph of y = 2x² - 11x - 9. The curve crosses the x-axis at x = 6.223111 and x = -0.723111; the vertex is at (2.75, -24.125).

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 = -11, c = -9.

    2x2−11x−9=02x^2 - 11x - 9 = 0
  2. 2

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

    D=(−11)2−4⋅2⋅(−9)=121+72=193D = (-11)^2 - 4\cdot2\cdot(-9) = 121 + 72 = 193
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=11±1934x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{193}}{4}
  5. 5

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

    x=11+1934orx=11−1934x = \frac{11 + \sqrt{193}}{4}\quad\text{or}\quad x = \frac{11 - \sqrt{193}}{4}
  6. 6

    As decimals: x ≈ 6.223111 and x ≈ -0.723111.

  7. 7

    Check numerically: for x ≈ 6.223111 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 11x - 9 = 0\ \Rightarrow\ x = \frac{11 + \sqrt{193}}{4},\ \frac{11 - \sqrt{193}}{4} \]
\[ 2x^2 - 11x - 9 = 0 \]
\[ D = (-11)^2 - 4\cdot2\cdot(-9) = 121 + 72 = 193 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{193}}{4} \]
\[ x = \frac{11 + \sqrt{193}}{4}\quad\text{or}\quad x = \frac{11 - \sqrt{193}}{4} \]

The parabola y = 2x² - 11x - 9

xy
-0.25-6.125
0.75-16.125
1.75-22.125
2.75-24.125
3.75-22.125
4.75-16.125
5.75-6.125

Questions

What are the solutions of 2x² - 11x - 9 = 0?

The solutions of 2x² - 11x - 9 = 0 are x = (11 + √193)/4 or x = (11 - √193)/4, that is x ≈ 6.223111 and x ≈ -0.723111.

Does 2x² - 11x - 9 = 0 factor?

Not over the integers, because the discriminant 193 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 11x - 9?

(11/4, -193/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 11/2, product = c/a = -9/2.

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    Solve 2x² - 11x - 9 = 0: x = (11 + √193)/4 or x = (11 - √193)/4 (steps) | WordGPT