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Solve -3x² + 5x - 2 = 0

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

x² +x += 0
−3x2+5x−2=0⇒x=23, 1-3x^2 + 5x - 2 = 0 \Rightarrow x = \frac{2}{3},\ 1
Discriminant D = 1 · x ≈ 0.666667, 1

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

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Graph of y = -3x² + 5x - 2-1123-20-15-10-5x = 0.66666667x = 1vertexy = -3x² + 5x - 2
Graph of y = -3x² + 5x - 2. The curve crosses the x-axis at x = 0.66666667 and x = 1; the vertex is at (0.83333333, 0.083333333).

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=1D = 5^2 - 4\cdot(-3)\cdot(-2) = 25 - 24 = 1
  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−2)(x−1)=0-\left(3x - 2\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−2=0⇒x=23x−1=0⇒x=13x - 2 = 0 \Rightarrow x = \frac{2}{3}\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−5±1−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{1}}{-6}
  7. 7

    Simplify to get the solutions:

    x=23orx=1x = \frac{2}{3}\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = 2/3 back in: a·x² + b·x + c = -3·(4/9) + 5·(2/3) + (-2) = 0 ✓

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

The parabola y = -3x² + 5x - 2

xy
-2.1667-26.917267
-1.1667-11.917067
-0.1667-2.9168667
0.83330.08333333
1.8333-2.9164667
2.8333-11.916267
3.8333-26.916067

Questions

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

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

Does -3x² + 5x - 2 = 0 factor?

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

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

(5/6, 1/12). 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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