WordGPT

Solve 3x² - 7x + 2 = 0

The solutions of 3x² - 7x + 2 = 0 are x = 2 or x = 1/3.

x² +x += 0
3x2−7x+2=0⇒x=2, 133x^2 - 7x + 2 = 0 \Rightarrow x = 2,\ \frac{1}{3}
Discriminant D = 25 · x ≈ 2, 0.333333

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 - 7x + 2-2-11234-551015202530x = 2x = 0.33333333vertexy = 3x² - 7x + 2
Graph of y = 3x² - 7x + 2. The curve crosses the x-axis at x = 2 and x = 0.33333333; the vertex is at (1.1666667, -2.0833333).

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 = -7, c = 2.

    3x2−7x+2=03x^2 - 7x + 2 = 0
  2. 2

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

    D=(−7)2−4⋅3⋅2=49−24=25D = (-7)^2 - 4\cdot3\cdot2 = 49 - 24 = 25
  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−2)(3x−1)=0\left(x - 2\right)\left(3x - 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:

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

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=7±256x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{25}}{6}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = 2 back in: a·x² + b·x + c = 3·(4) + (-7)·(2) + 2 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 7x + 2 = 0\ \Rightarrow\ x = 2,\ \frac{1}{3} \]
\[ 3x^2 - 7x + 2 = 0 \]
\[ D = (-7)^2 - 4\cdot3\cdot2 = 49 - 24 = 25 \]
\[ \left(x - 2\right)\left(3x - 1\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad 3x - 1 = 0 \Rightarrow x = \frac{1}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{25}}{6} \]
\[ x = 2\quad\text{or}\quad x = \frac{1}{3} \]

The parabola y = 3x² - 7x + 2

xy
-1.833324.916067
-0.83339.9162667
0.16670.91646667
1.1667-2.0833333
2.16670.91686667
3.16679.9170667
4.166724.917267

Questions

What are the solutions of 3x² - 7x + 2 = 0?

The solutions of 3x² - 7x + 2 = 0 are x = 2 or x = 1/3.

Does 3x² - 7x + 2 = 0 factor?

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

What is the vertex of y = 3x² - 7x + 2?

(7/6, -25/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 7/3, product = c/a = 2/3.

Similar equations

    Solve 3x² - 7x + 2 = 0: x = 2 or x = 1/3 (steps) | WordGPT