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Solve -3x² + 5x + 2 = 0

The solutions of -3x² + 5x + 2 = 0 are x = -1/3 or x = 2.

x² +x += 0
−3x2+5x+2=0⇒x=−13, 2-3x^2 + 5x + 2 = 0 \Rightarrow x = -\frac{1}{3},\ 2
Discriminant D = 49 · x ≈ -0.333333, 2

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

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Graph of y = -3x² + 5x + 2-2-11234-30-20-10x = -0.33333333x = 2vertexy = -3x² + 5x + 2
Graph of y = -3x² + 5x + 2. The curve crosses the x-axis at x = -0.33333333 and x = 2; the vertex is at (0.83333333, 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 = 5, c = 2.

    −3x2+5x+2=0-3x^2 + 5x + 2 = 0
  2. 2

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

    D=52−4⋅(−3)⋅2=25+24=49D = 5^2 - 4\cdot(-3)\cdot2 = 25 + 24 = 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+1)(x−2)=0-\left(3x + 1\right)\left(x - 2\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−2=0⇒x=23x + 1 = 0 \Rightarrow x = -\frac{1}{3}\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−5±49−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{49}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−13orx=2x = -\frac{1}{3}\quad\text{or}\quad x = 2
  8. 8

    Check by putting x = -1/3 back in: a·x² + b·x + c = -3·(1/9) + 5·(-1/3) + 2 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 5x + 2 = 0\ \Rightarrow\ x = -\frac{1}{3},\ 2 \]
\[ -3x^2 + 5x + 2 = 0 \]
\[ D = 5^2 - 4\cdot(-3)\cdot2 = 25 + 24 = 49 \]
\[ -\left(3x + 1\right)\left(x - 2\right) = 0 \]
\[ 3x + 1 = 0 \Rightarrow x = -\frac{1}{3}\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{49}}{-6} \]
\[ x = -\frac{1}{3}\quad\text{or}\quad x = 2 \]

The parabola y = -3x² + 5x + 2

xy
-2.1667-22.917267
-1.1667-7.9170667
-0.16671.0831333
0.83334.0833333
1.83331.0835333
2.8333-7.9162667
3.8333-22.916067

Questions

What are the solutions of -3x² + 5x + 2 = 0?

The solutions of -3x² + 5x + 2 = 0 are x = -1/3 or x = 2.

Does -3x² + 5x + 2 = 0 factor?

Yes: -1(3x+1)(x-2) = 0.

What is the vertex of y = -3x² + 5x + 2?

(5/6, 49/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 5/3, product = c/a = -2/3.

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