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Solve 2x² - 11x + 9 = 0

The solutions of 2x² - 11x + 9 = 0 are x = 9/2 or x = 1.

x² +x += 0
2x2−11x+9=0⇒x=92, 12x^2 - 11x + 9 = 0 \Rightarrow x = \frac{9}{2},\ 1
Discriminant D = 49 · x ≈ 4.5, 1

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

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Graph of y = 2x² - 11x + 9-11234567-10-551015202530x = 4.5x = 1vertexy = 2x² - 11x + 9
Graph of y = 2x² - 11x + 9. The curve crosses the x-axis at x = 4.5 and x = 1; the vertex is at (2.75, -6.125).

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=49D = (-11)^2 - 4\cdot2\cdot9 = 121 - 72 = 49
  3. 3

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

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    (2x−9)(x−1)=0\left(2x - 9\right)\left(x - 1\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    2x−9=0⇒x=92x−1=0⇒x=12x - 9 = 0 \Rightarrow x = \frac{9}{2}\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=11±494x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{49}}{4}
  7. 7

    Simplify to get the solutions:

    x=92orx=1x = \frac{9}{2}\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = 9/2 back in: a·x² + b·x + c = 2·(81/4) + (-11)·(9/2) + 9 = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 11x + 9 = 0\ \Rightarrow\ x = \frac{9}{2},\ 1 \]
\[ 2x^2 - 11x + 9 = 0 \]
\[ D = (-11)^2 - 4\cdot2\cdot9 = 121 - 72 = 49 \]
\[ \left(2x - 9\right)\left(x - 1\right) = 0 \]
\[ 2x - 9 = 0 \Rightarrow x = \frac{9}{2}\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{49}}{4} \]
\[ x = \frac{9}{2}\quad\text{or}\quad x = 1 \]

The parabola y = 2x² - 11x + 9

xy
-0.2511.875
0.751.875
1.75-4.125
2.75-6.125
3.75-4.125
4.751.875
5.7511.875

Questions

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

The solutions of 2x² - 11x + 9 = 0 are x = 9/2 or x = 1.

Does 2x² - 11x + 9 = 0 factor?

Yes: (2x-9)(x-1) = 0.

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

(11/4, -49/8). 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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