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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, −23x^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 - 2-4-3-2-112102030x = 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=03x^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\cdot3\cdot(-2) = 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±496x = \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
\[ 3x^2 + 5x - 2 = 0\ \Rightarrow\ x = \frac{1}{3},\ -2 \]
\[ 3x^2 + 5x - 2 = 0 \]
\[ D = 5^2 - 4\cdot3\cdot(-2) = 25 + 24 = 49 \]
\[ \left(3x - 1\right)\left(x + 2\right) = 0 \]
\[ 3x - 1 = 0 \Rightarrow x = \frac{1}{3}\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{49}}{6} \]
\[ x = \frac{1}{3}\quad\text{or}\quad x = -2 \]

The parabola y = 3x² + 5x - 2

xy
-3.833322.916067
-2.83337.9162667
-1.8333-1.0835333
-0.8333-4.0833333
0.1667-1.0831333
1.16677.9170667
2.166722.917267

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

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

(-5/6, -49/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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