WordGPT

Solve 2x² + 5x + 2 = 0

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

x² +x += 0
2x2+5x+2=0⇒x=−12, −22x^2 + 5x + 2 = 0 \Rightarrow x = -\frac{1}{2},\ -2
Discriminant D = 9 · x ≈ -0.5, -2

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 + 5x + 2-4-3-2-1125101520x = -0.5x = -2vertexy = 2x² + 5x + 2
Graph of y = 2x² + 5x + 2. The curve crosses the x-axis at x = -0.5 and x = -2; the vertex is at (-1.25, -1.125).

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 = 5, c = 2.

    2x2+5x+2=02x^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⋅2⋅2=25−16=9D = 5^2 - 4\cdot2\cdot2 = 25 - 16 = 9
  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+1)(x+2)=0\left(2x + 1\right)\left(x + 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:

    2x+1=0⇒x=−12x+2=0⇒x=−22x + 1 = 0 \Rightarrow x = -\frac{1}{2}\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−5±94x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{9}}{4}
  7. 7

    Simplify to get the solutions:

    x=−12orx=−2x = -\frac{1}{2}\quad\text{or}\quad x = -2
  8. 8

    Check by putting x = -1/2 back in: a·x² + b·x + c = 2·(1/4) + 5·(-1/2) + 2 = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 5x + 2 = 0\ \Rightarrow\ x = -\frac{1}{2},\ -2 \]
\[ 2x^2 + 5x + 2 = 0 \]
\[ D = 5^2 - 4\cdot2\cdot2 = 25 - 16 = 9 \]
\[ \left(2x + 1\right)\left(x + 2\right) = 0 \]
\[ 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{9}}{4} \]
\[ x = -\frac{1}{2}\quad\text{or}\quad x = -2 \]

The parabola y = 2x² + 5x + 2

xy
-4.2516.875
-3.256.875
-2.250.875
-1.25-1.125
-0.250.875
0.756.875
1.7516.875

Questions

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

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

Does 2x² + 5x + 2 = 0 factor?

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

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

(-5/4, -9/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -5/2, product = c/a = 1.

Similar equations

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