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Solve -3x² - x + 4 = 0

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

x² +x += 0
−3x2−x+4=0⇒x=−43, 1-3x^2 - x + 4 = 0 \Rightarrow x = -\frac{4}{3},\ 1
Discriminant D = 49 · x ≈ -1.33333, 1

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

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Graph of y = -3x² - x + 4-3-2-1123-30-20-10x = -1.3333333x = 1vertexy = -3x² - x + 4
Graph of y = -3x² - x + 4. The curve crosses the x-axis at x = -1.3333333 and x = 1; 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=0-3x^2 - x + 4 = 0
  2. 2

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

    D=(−1)2−4⋅(−3)⋅4=1+48=49D = (-1)^2 - 4\cdot(-3)\cdot4 = 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:

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

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

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=1±49−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{49}}{-6}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -4/3 back in: a·x² + b·x + c = -3·(16/9) + (-1)·(-4/3) + 4 = 0 ✓

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

The parabola y = -3x² - x + 4

xy
-3.1667-22.917267
-2.1667-7.9170667
-1.16671.0831333
-0.16674.0833333
0.83331.0835333
1.8333-7.9162667
2.8333-22.916067

Questions

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

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

Does -3x² - x + 4 = 0 factor?

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

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

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

What are the sum and product of the roots?

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

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