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Solve 3x² + 7x + 4 = 0

The solutions of 3x² + 7x + 4 = 0 are x = -1 or x = -4/3.

x² +x += 0
3x2+7x+4=0⇒x=−1, −433x^2 + 7x + 4 = 0 \Rightarrow x = -1,\ -\frac{4}{3}
Discriminant D = 1 · x ≈ -1, -1.33333

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

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Graph of y = 3x² + 7x + 4-3-2-115101520x = -1x = -1.3333333vertexy = 3x² + 7x + 4
Graph of y = 3x² + 7x + 4. The curve crosses the x-axis at x = -1 and x = -1.3333333; the vertex is at (-1.1666667, -0.083333333).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 3, b = 7, c = 4.

    3x2+7x+4=03x^2 + 7x + 4 = 0
  2. 2

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

    D=72−4⋅3⋅4=49−48=1D = 7^2 - 4\cdot3\cdot4 = 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:

    (x+1)(3x+4)=0\left(x + 1\right)\left(3x + 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:

    x+1=0⇒x=−13x+4=0⇒x=−43x + 1 = 0 \Rightarrow x = -1\qquad 3x + 4 = 0 \Rightarrow x = -\frac{4}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−7±16x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{1}}{6}
  7. 7

    Simplify to get the solutions:

    x=−1orx=−43x = -1\quad\text{or}\quad x = -\frac{4}{3}
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = 3·(1) + 7·(-1) + 4 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 7x + 4 = 0\ \Rightarrow\ x = -1,\ -\frac{4}{3} \]
\[ 3x^2 + 7x + 4 = 0 \]
\[ D = 7^2 - 4\cdot3\cdot4 = 49 - 48 = 1 \]
\[ \left(x + 1\right)\left(3x + 4\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 3x + 4 = 0 \Rightarrow x = -\frac{4}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{1}}{6} \]
\[ x = -1\quad\text{or}\quad x = -\frac{4}{3} \]

The parabola y = 3x² + 7x + 4

xy
-4.166726.917267
-3.166711.917067
-2.16672.9168667
-1.1667-0.08333333
-0.16672.9164667
0.833311.916267
1.833326.916067

Questions

What are the solutions of 3x² + 7x + 4 = 0?

The solutions of 3x² + 7x + 4 = 0 are x = -1 or x = -4/3.

Does 3x² + 7x + 4 = 0 factor?

Yes: (x+1)(3x+4) = 0.

What is the vertex of y = 3x² + 7x + 4?

(-7/6, -1/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -7/3, product = c/a = 4/3.

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