WordGPT

Solve 3x² - 5x + 2 = 0

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

x² +x += 0
3x2−5x+2=0⇒x=1, 233x^2 - 5x + 2 = 0 \Rightarrow x = 1,\ \frac{2}{3}
Discriminant D = 1 · x ≈ 1, 0.666667

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

Explain it step by step in chatMake a short video of it
Graph of y = 3x² - 5x + 2-11235101520x = 1x = 0.66666667vertexy = 3x² - 5x + 2
Graph of y = 3x² - 5x + 2. The curve crosses the x-axis at x = 1 and x = 0.66666667; 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=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=(−5)2−4⋅3⋅2=25−24=1D = (-5)^2 - 4\cdot3\cdot2 = 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:

    (x−1)(3x−2)=0\left(x - 1\right)\left(3x - 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:

    x−1=0⇒x=13x−2=0⇒x=23x - 1 = 0 \Rightarrow x = 1\qquad 3x - 2 = 0 \Rightarrow x = \frac{2}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=5±16x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{1}}{6}
  7. 7

    Simplify to get the solutions:

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

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

LaTeX source of the formulas on this page
\[ 3x^2 - 5x + 2 = 0\ \Rightarrow\ x = 1,\ \frac{2}{3} \]
\[ 3x^2 - 5x + 2 = 0 \]
\[ D = (-5)^2 - 4\cdot3\cdot2 = 25 - 24 = 1 \]
\[ \left(x - 1\right)\left(3x - 2\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 3x - 2 = 0 \Rightarrow x = \frac{2}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{1}}{6} \]
\[ x = 1\quad\text{or}\quad x = \frac{2}{3} \]

The parabola y = 3x² - 5x + 2

xy
-2.166726.917267
-1.166711.917067
-0.16672.9168667
0.8333-0.08333333
1.83332.9164667
2.833311.916267
3.833326.916067

Questions

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

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

Does 3x² - 5x + 2 = 0 factor?

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

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

(5/6, -1/12). 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.

Similar equations

    Solve 3x² - 5x + 2 = 0: x = 1 or x = 2/3 (steps) | WordGPT