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Solve -3x² + 11x + 4 = 0

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

x² +x += 0
−3x2+11x+4=0⇒x=−13, 4-3x^2 + 11x + 4 = 0 \Rightarrow x = -\frac{1}{3},\ 4
Discriminant D = 169 · x ≈ -0.333333, 4

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

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Graph of y = -3x² + 11x + 4-2-1123456-50-40-30-20-101020x = -0.33333333x = 4vertexy = -3x² + 11x + 4
Graph of y = -3x² + 11x + 4. The curve crosses the x-axis at x = -0.33333333 and x = 4; 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=0-3x^2 + 11x + 4 = 0
  2. 2

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

    D=112−4⋅(−3)⋅4=121+48=169D = 11^2 - 4\cdot(-3)\cdot4 = 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:

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

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

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

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

    Check by putting x = -1/3 back in: a·x² + b·x + c = -3·(1/9) + 11·(-1/3) + 4 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 11x + 4 = 0\ \Rightarrow\ x = -\frac{1}{3},\ 4 \]
\[ -3x^2 + 11x + 4 = 0 \]
\[ D = 11^2 - 4\cdot(-3)\cdot4 = 121 + 48 = 169 \]
\[ -\left(3x + 1\right)\left(x - 4\right) = 0 \]
\[ 3x + 1 = 0 \Rightarrow x = -\frac{1}{3}\qquad x - 4 = 0 \Rightarrow x = 4 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{169}}{-6} \]
\[ x = -\frac{1}{3}\quad\text{or}\quad x = 4 \]

The parabola y = -3x² + 11x + 4

xy
-1.1667-12.917267
-0.16672.0829333
0.833311.083133
1.833314.083333
2.833311.083533
3.83332.0837333
4.8333-12.916067

Questions

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

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

Does -3x² + 11x + 4 = 0 factor?

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

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

(11/6, 169/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

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

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