WordGPT

Solve -3x² - 5x + 2 = 0

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

x² +x += 0
−3x2−5x+2=0⇒x=−2, 13-3x^2 - 5x + 2 = 0 \Rightarrow x = -2,\ \frac{1}{3}
Discriminant D = 49 · 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 - 5x + 2-4-3-2-112-30-20-10x = -2x = 0.33333333vertexy = -3x² - 5x + 2
Graph of y = -3x² - 5x + 2. The curve crosses the x-axis at x = -2 and x = 0.33333333; 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=(−5)2−4⋅(−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:

    −(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=5±49−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{49}}{-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) + (-5)·(-2) + 2 = 0 ✓

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

The parabola y = -3x² - 5x + 2

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

Questions

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

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

Does -3x² - 5x + 2 = 0 factor?

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

Similar equations

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