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Solve 2x² + 11x + 12 = 0

The solutions of 2x² + 11x + 12 = 0 are x = -3/2 or x = -4.

x² +x += 0
2x2+11x+12=0⇒x=−32, −42x^2 + 11x + 12 = 0 \Rightarrow x = -\frac{3}{2},\ -4
Discriminant D = 25 · x ≈ -1.5, -4

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

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Graph of y = 2x² + 11x + 12-6-5-4-3-2-11-5510152025x = -1.5x = -4vertexy = 2x² + 11x + 12
Graph of y = 2x² + 11x + 12. The curve crosses the x-axis at x = -1.5 and x = -4; the vertex is at (-2.75, -3.125).

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 = 12.

    2x2+11x+12=02x^2 + 11x + 12 = 0
  2. 2

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

    D=112−4⋅2⋅12=121−96=25D = 11^2 - 4\cdot2\cdot12 = 121 - 96 = 25
  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+4)=0\left(2x + 3\right)\left(x + 4\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+4=0⇒x=−42x + 3 = 0 \Rightarrow x = -\frac{3}{2}\qquad x + 4 = 0 \Rightarrow x = -4
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−11±254x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{25}}{4}
  7. 7

    Simplify to get the solutions:

    x=−32orx=−4x = -\frac{3}{2}\quad\text{or}\quad x = -4
  8. 8

    Check by putting x = -3/2 back in: a·x² + b·x + c = 2·(9/4) + 11·(-3/2) + 12 = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 11x + 12 = 0\ \Rightarrow\ x = -\frac{3}{2},\ -4 \]
\[ 2x^2 + 11x + 12 = 0 \]
\[ D = 11^2 - 4\cdot2\cdot12 = 121 - 96 = 25 \]
\[ \left(2x + 3\right)\left(x + 4\right) = 0 \]
\[ 2x + 3 = 0 \Rightarrow x = -\frac{3}{2}\qquad x + 4 = 0 \Rightarrow x = -4 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{25}}{4} \]
\[ x = -\frac{3}{2}\quad\text{or}\quad x = -4 \]

The parabola y = 2x² + 11x + 12

xy
-5.7514.875
-4.754.875
-3.75-1.125
-2.75-3.125
-1.75-1.125
-0.754.875
0.2514.875

Questions

What are the solutions of 2x² + 11x + 12 = 0?

The solutions of 2x² + 11x + 12 = 0 are x = -3/2 or x = -4.

Does 2x² + 11x + 12 = 0 factor?

Yes: (2x+3)(x+4) = 0.

What is the vertex of y = 2x² + 11x + 12?

(-11/4, -25/8). 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 = 6.

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