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Solve 2x² - x - 6 = 0

The solutions of 2x² - x - 6 = 0 are x = 2 or x = -3/2.

x² +x += 0
2x2−x−6=0⇒x=2, −322x^2 - x - 6 = 0 \Rightarrow x = 2,\ -\frac{3}{2}
Discriminant D = 49 · x ≈ 2, -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 - x - 6-4-3-2-11234-10-551015202530x = 2x = -1.5vertexy = 2x² - x - 6
Graph of y = 2x² - x - 6. The curve crosses the x-axis at x = 2 and x = -1.5; the vertex is at (0.25, -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 = -1, c = -6.

    2x2−x−6=02x^2 - x - 6 = 0
  2. 2

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

    D=(−1)2−4⋅2⋅(−6)=1+48=49D = (-1)^2 - 4\cdot2\cdot(-6) = 1 + 48 = 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:

    (x−2)(2x+3)=0\left(x - 2\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−2=0⇒x=22x+3=0⇒x=−32x - 2 = 0 \Rightarrow x = 2\qquad 2x + 3 = 0 \Rightarrow x = -\frac{3}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=1±494x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{49}}{4}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = 2 back in: a·x² + b·x + c = 2·(4) + (-1)·(2) + (-6) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 - x - 6 = 0\ \Rightarrow\ x = 2,\ -\frac{3}{2} \]
\[ 2x^2 - x - 6 = 0 \]
\[ D = (-1)^2 - 4\cdot2\cdot(-6) = 1 + 48 = 49 \]
\[ \left(x - 2\right)\left(2x + 3\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad 2x + 3 = 0 \Rightarrow x = -\frac{3}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{49}}{4} \]
\[ x = 2\quad\text{or}\quad x = -\frac{3}{2} \]

The parabola y = 2x² - x - 6

xy
-2.7511.875
-1.751.875
-0.75-4.125
0.25-6.125
1.25-4.125
2.251.875
3.2511.875

Questions

What are the solutions of 2x² - x - 6 = 0?

The solutions of 2x² - x - 6 = 0 are x = 2 or x = -3/2.

Does 2x² - x - 6 = 0 factor?

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

What is the vertex of y = 2x² - x - 6?

(1/4, -49/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1/2, product = c/a = -3.

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