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Solve 3x² + x - 4 = 0

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

x² +x += 0
3x2+x−4=0⇒x=1, −433x^2 + x - 4 = 0 \Rightarrow x = 1,\ -\frac{4}{3}
Discriminant D = 49 · 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 + x - 4-3-2-1123102030x = 1x = -1.3333333vertexy = 3x² + x - 4
Graph of y = 3x² + x - 4. The curve crosses the x-axis at x = 1 and x = -1.3333333; the vertex is at (-0.16666667, -4.0833333).

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 = 1, c = -4.

    3x2+x−4=03x^2 + x - 4 = 0
  2. 2

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

    D=12−4⋅3⋅(−4)=1+48=49D = 1^2 - 4\cdot3\cdot(-4) = 1 + 48 = 49
  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=−1±496x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{49}}{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) + 1·(1) + (-4) = 0 ✓

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

The parabola y = 3x² + x - 4

xy
-3.166722.917267
-2.16677.9170667
-1.1667-1.0831333
-0.1667-4.0833333
0.8333-1.0835333
1.83337.9162667
2.833322.916067

Questions

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

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

Does 3x² + x - 4 = 0 factor?

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

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

(-1/6, -49/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -1/3, product = c/a = -4/3.

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