WordGPT

Solve -2x² - x + 3 = 0

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

x² +x += 0
−2x2−x+3=0⇒x=−32, 1-2x^2 - x + 3 = 0 \Rightarrow x = -\frac{3}{2},\ 1
Discriminant D = 25 · x ≈ -1.5, 1

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 = -2x² - x + 3-4-3-2-1123-25-20-15-10-55x = -1.5x = 1vertexy = -2x² - x + 3
Graph of y = -2x² - x + 3. The curve crosses the x-axis at x = -1.5 and x = 1; the vertex is at (-0.25, 3.125).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -2, b = -1, c = 3.

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

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

    D=(−1)2−4⋅(−2)⋅3=1+24=25D = (-1)^2 - 4\cdot(-2)\cdot3 = 1 + 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:

    −(2x+3)(x−1)=0-\left(2x + 3\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:

    2x+3=0⇒x=−32x−1=0⇒x=12x + 3 = 0 \Rightarrow x = -\frac{3}{2}\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=1±25−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{25}}{-4}
  7. 7

    Simplify to get the solutions:

    x=−32orx=1x = -\frac{3}{2}\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = -3/2 back in: a·x² + b·x + c = -2·(9/4) + (-1)·(-3/2) + 3 = 0 ✓

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

The parabola y = -2x² - x + 3

xy
-3.25-14.875
-2.25-4.875
-1.251.125
-0.253.125
0.751.125
1.75-4.875
2.75-14.875

Questions

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

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

Does -2x² - x + 3 = 0 factor?

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

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

(-1/4, 25/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -1/2, product = c/a = -3/2.

Similar equations

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