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Solve 2x² - 3x - 9 = 0

The solutions of 2x² - 3x - 9 = 0 are x = 3 or x = -3/2.

x² +x += 0
2x2−3x−9=0⇒x=3, −322x^2 - 3x - 9 = 0 \Rightarrow x = 3,\ -\frac{3}{2}
Discriminant D = 81 · x ≈ 3, -1.5

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

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Graph of y = 2x² - 3x - 9-4-3-2-112345-10102030x = 3x = -1.5vertexy = 2x² - 3x - 9
Graph of y = 2x² - 3x - 9. The curve crosses the x-axis at x = 3 and x = -1.5; the vertex is at (0.75, -10.125).

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 = -9.

    2x2−3x−9=02x^2 - 3x - 9 = 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⋅(−9)=9+72=81D = (-3)^2 - 4\cdot2\cdot(-9) = 9 + 72 = 81
  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:

    (x−3)(2x+3)=0\left(x - 3\right)\left(2x + 3\right) = 0
  5. 5

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

    x−3=0⇒x=32x+3=0⇒x=−32x - 3 = 0 \Rightarrow x = 3\qquad 2x + 3 = 0 \Rightarrow x = -\frac{3}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=3±814x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{81}}{4}
  7. 7

    Simplify to get the solutions:

    x=3orx=−32x = 3\quad\text{or}\quad x = -\frac{3}{2}
  8. 8

    Check by putting x = 3 back in: a·x² + b·x + c = 2·(9) + (-3)·(3) + (-9) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 3x - 9 = 0\ \Rightarrow\ x = 3,\ -\frac{3}{2} \]
\[ 2x^2 - 3x - 9 = 0 \]
\[ D = (-3)^2 - 4\cdot2\cdot(-9) = 9 + 72 = 81 \]
\[ \left(x - 3\right)\left(2x + 3\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad 2x + 3 = 0 \Rightarrow x = -\frac{3}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{81}}{4} \]
\[ x = 3\quad\text{or}\quad x = -\frac{3}{2} \]

The parabola y = 2x² - 3x - 9

xy
-2.257.875
-1.25-2.125
-0.25-8.125
0.75-10.125
1.75-8.125
2.75-2.125
3.757.875

Questions

What are the solutions of 2x² - 3x - 9 = 0?

The solutions of 2x² - 3x - 9 = 0 are x = 3 or x = -3/2.

Does 2x² - 3x - 9 = 0 factor?

Yes: (x-3)(2x+3) = 0.

What is the vertex of y = 2x² - 3x - 9?

(3/4, -81/8). 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 = -9/2.

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