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Solve 5x² + 12x + 4 = 0

The solutions of 5x² + 12x + 4 = 0 are x = -2/5 or x = -2.

x² +x += 0
5x2+12x+4=0⇒x=−25, −25x^2 + 12x + 4 = 0 \Rightarrow x = -\frac{2}{5},\ -2
Discriminant D = 64 · x ≈ -0.4, -2

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

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Graph of y = 5x² + 12x + 4-4-3-2-112-101020304050x = -0.4x = -2vertexy = 5x² + 12x + 4
Graph of y = 5x² + 12x + 4. The curve crosses the x-axis at x = -0.4 and x = -2; the vertex is at (-1.2, -3.2).

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 = 12, c = 4.

    5x2+12x+4=05x^2 + 12x + 4 = 0
  2. 2

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

    D=122−4⋅5⋅4=144−80=64D = 12^2 - 4\cdot5\cdot4 = 144 - 80 = 64
  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+2)(x+2)=0\left(5x + 2\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+2=0⇒x=−25x+2=0⇒x=−25x + 2 = 0 \Rightarrow x = -\frac{2}{5}\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−12±6410x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{64}}{10}
  7. 7

    Simplify to get the solutions:

    x=−25orx=−2x = -\frac{2}{5}\quad\text{or}\quad x = -2
  8. 8

    Check by putting x = -2/5 back in: a·x² + b·x + c = 5·(4/25) + 12·(-2/5) + 4 = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 + 12x + 4 = 0\ \Rightarrow\ x = -\frac{2}{5},\ -2 \]
\[ 5x^2 + 12x + 4 = 0 \]
\[ D = 12^2 - 4\cdot5\cdot4 = 144 - 80 = 64 \]
\[ \left(5x + 2\right)\left(x + 2\right) = 0 \]
\[ 5x + 2 = 0 \Rightarrow x = -\frac{2}{5}\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{64}}{10} \]
\[ x = -\frac{2}{5}\quad\text{or}\quad x = -2 \]

The parabola y = 5x² + 12x + 4

xy
-4.241.8
-3.216.8
-2.21.8
-1.2-3.2
-0.21.8
0.816.8
1.841.8

Questions

What are the solutions of 5x² + 12x + 4 = 0?

The solutions of 5x² + 12x + 4 = 0 are x = -2/5 or x = -2.

Does 5x² + 12x + 4 = 0 factor?

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

What is the vertex of y = 5x² + 12x + 4?

(-6/5, -16/5). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -12/5, product = c/a = 4/5.

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