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Solve 5x² + 11x + 2 = 0

The solutions of 5x² + 11x + 2 = 0 are x = -1/5 or x = -2.

x² +x += 0
5x2+11x+2=0⇒x=−15, −25x^2 + 11x + 2 = 0 \Rightarrow x = -\frac{1}{5},\ -2
Discriminant D = 81 · x ≈ -0.2, -2

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

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Graph of y = 5x² + 11x + 2-4-3-2-112-101020304050x = -0.2x = -2vertexy = 5x² + 11x + 2
Graph of y = 5x² + 11x + 2. The curve crosses the x-axis at x = -0.2 and x = -2; the vertex is at (-1.1, -4.05).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = 11, c = 2.

    5x2+11x+2=05x^2 + 11x + 2 = 0
  2. 2

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

    D=112−4⋅5⋅2=121−40=81D = 11^2 - 4\cdot5\cdot2 = 121 - 40 = 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:

    (5x+1)(x+2)=0\left(5x + 1\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:

    5x+1=0⇒x=−15x+2=0⇒x=−25x + 1 = 0 \Rightarrow x = -\frac{1}{5}\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−11±8110x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{81}}{10}
  7. 7

    Simplify to get the solutions:

    x=−15orx=−2x = -\frac{1}{5}\quad\text{or}\quad x = -2
  8. 8

    Check by putting x = -1/5 back in: a·x² + b·x + c = 5·(1/25) + 11·(-1/5) + 2 = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 + 11x + 2 = 0\ \Rightarrow\ x = -\frac{1}{5},\ -2 \]
\[ 5x^2 + 11x + 2 = 0 \]
\[ D = 11^2 - 4\cdot5\cdot2 = 121 - 40 = 81 \]
\[ \left(5x + 1\right)\left(x + 2\right) = 0 \]
\[ 5x + 1 = 0 \Rightarrow x = -\frac{1}{5}\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{81}}{10} \]
\[ x = -\frac{1}{5}\quad\text{or}\quad x = -2 \]

The parabola y = 5x² + 11x + 2

xy
-4.140.95
-3.115.95
-2.10.95
-1.1-4.05
-0.10.95
0.915.95
1.940.95

Questions

What are the solutions of 5x² + 11x + 2 = 0?

The solutions of 5x² + 11x + 2 = 0 are x = -1/5 or x = -2.

Does 5x² + 11x + 2 = 0 factor?

Yes: (5x+1)(x+2) = 0.

What is the vertex of y = 5x² + 11x + 2?

(-11/10, -81/20). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -11/5, product = c/a = 2/5.

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