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Solve 2x² + x - 6 = 0

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

x² +x += 0
2x2+x−6=0⇒x=32, −22x^2 + x - 6 = 0 \Rightarrow x = \frac{3}{2},\ -2
Discriminant D = 49 · x ≈ 1.5, -2

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 = 1.5x = -2vertexy = 2x² + x - 6
Graph of y = 2x² + x - 6. The curve crosses the x-axis at x = 1.5 and x = -2; 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=12−4⋅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:

    (2x−3)(x+2)=0\left(2x - 3\right)\left(x + 2\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−3=0⇒x=32x+2=0⇒x=−22x - 3 = 0 \Rightarrow x = \frac{3}{2}\qquad x + 2 = 0 \Rightarrow x = -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=32orx=−2x = \frac{3}{2}\quad\text{or}\quad x = -2
  8. 8

    Check by putting x = 3/2 back in: a·x² + b·x + c = 2·(9/4) + 1·(3/2) + (-6) = 0 ✓

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

The parabola y = 2x² + x - 6

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

Questions

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

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

Does 2x² + x - 6 = 0 factor?

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