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Solve 3x² - 11x - 4 = 0

The solutions of 3x² - 11x - 4 = 0 are x = 4 or x = -1/3.

x² +x += 0
3x2−11x−4=0⇒x=4, −133x^2 - 11x - 4 = 0 \Rightarrow x = 4,\ -\frac{1}{3}
Discriminant D = 169 · x ≈ 4, -0.333333

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

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Graph of y = 3x² - 11x - 4-2-1123456-20-101020304050x = 4x = -0.33333333vertexy = 3x² - 11x - 4
Graph of y = 3x² - 11x - 4. The curve crosses the x-axis at x = 4 and x = -0.33333333; the vertex is at (1.8333333, -14.083333).

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 = -11, c = -4.

    3x2−11x−4=03x^2 - 11x - 4 = 0
  2. 2

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

    D=(−11)2−4⋅3⋅(−4)=121+48=169D = (-11)^2 - 4\cdot3\cdot(-4) = 121 + 48 = 169
  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−4)(3x+1)=0\left(x - 4\right)\left(3x + 1\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−4=0⇒x=43x+1=0⇒x=−13x - 4 = 0 \Rightarrow x = 4\qquad 3x + 1 = 0 \Rightarrow x = -\frac{1}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=11±1696x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{169}}{6}
  7. 7

    Simplify to get the solutions:

    x=4orx=−13x = 4\quad\text{or}\quad x = -\frac{1}{3}
  8. 8

    Check by putting x = 4 back in: a·x² + b·x + c = 3·(16) + (-11)·(4) + (-4) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 11x - 4 = 0\ \Rightarrow\ x = 4,\ -\frac{1}{3} \]
\[ 3x^2 - 11x - 4 = 0 \]
\[ D = (-11)^2 - 4\cdot3\cdot(-4) = 121 + 48 = 169 \]
\[ \left(x - 4\right)\left(3x + 1\right) = 0 \]
\[ x - 4 = 0 \Rightarrow x = 4\qquad 3x + 1 = 0 \Rightarrow x = -\frac{1}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{169}}{6} \]
\[ x = 4\quad\text{or}\quad x = -\frac{1}{3} \]

The parabola y = 3x² - 11x - 4

xy
-1.166712.917267
-0.1667-2.0829333
0.8333-11.083133
1.8333-14.083333
2.8333-11.083533
3.8333-2.0837333
4.833312.916067

Questions

What are the solutions of 3x² - 11x - 4 = 0?

The solutions of 3x² - 11x - 4 = 0 are x = 4 or x = -1/3.

Does 3x² - 11x - 4 = 0 factor?

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

What is the vertex of y = 3x² - 11x - 4?

(11/6, -169/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 11/3, product = c/a = -4/3.

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