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Solve 2x² + 7x + 6 = 0

The solutions of 2x² + 7x + 6 = 0 are x = -3/2 or x = -2.

x² +x += 0
2x2+7x+6=0⇒x=−32, −22x^2 + 7x + 6 = 0 \Rightarrow x = -\frac{3}{2},\ -2
Discriminant D = 1 · 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 + 7x + 6-4-3-2-11-2246810121416x = -1.5x = -2vertexy = 2x² + 7x + 6
Graph of y = 2x² + 7x + 6. The curve crosses the x-axis at x = -1.5 and x = -2; the vertex is at (-1.75, -0.125).

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 = 7, c = 6.

    2x2+7x+6=02x^2 + 7x + 6 = 0
  2. 2

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

    D=72−4⋅2⋅6=49−48=1D = 7^2 - 4\cdot2\cdot6 = 49 - 48 = 1
  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=−7±14x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{1}}{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) + 7·(-3/2) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 7x + 6 = 0\ \Rightarrow\ x = -\frac{3}{2},\ -2 \]
\[ 2x^2 + 7x + 6 = 0 \]
\[ D = 7^2 - 4\cdot2\cdot6 = 49 - 48 = 1 \]
\[ \left(2x + 3\right)\left(x + 2\right) = 0 \]
\[ 2x + 3 = 0 \Rightarrow x = -\frac{3}{2}\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{1}}{4} \]
\[ x = -\frac{3}{2}\quad\text{or}\quad x = -2 \]

The parabola y = 2x² + 7x + 6

xy
-4.7517.875
-3.757.875
-2.751.875
-1.75-0.125
-0.751.875
0.257.875
1.2517.875

Questions

What are the solutions of 2x² + 7x + 6 = 0?

The solutions of 2x² + 7x + 6 = 0 are x = -3/2 or x = -2.

Does 2x² + 7x + 6 = 0 factor?

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

What is the vertex of y = 2x² + 7x + 6?

(-7/4, -1/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -7/2, product = c/a = 3.

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