WordGPT

Solve 2x² + 9x + 10 = 0

The solutions of 2x² + 9x + 10 = 0 are x = -2 or x = -5/2.

x² +x += 0
2x2+9x+10=0⇒x=−2, −522x^2 + 9x + 10 = 0 \Rightarrow x = -2,\ -\frac{5}{2}
Discriminant D = 1 · x ≈ -2, -2.5

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 + 9x + 10-5-4-3-2-1-2246810121416x = -2x = -2.5vertexy = 2x² + 9x + 10
Graph of y = 2x² + 9x + 10. The curve crosses the x-axis at x = -2 and x = -2.5; the vertex is at (-2.25, -0.125).

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 = 9, c = 10.

    2x2+9x+10=02x^2 + 9x + 10 = 0
  2. 2

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

    D=92−4⋅2⋅10=81−80=1D = 9^2 - 4\cdot2\cdot10 = 81 - 80 = 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+2)(2x+5)=0\left(x + 2\right)\left(2x + 5\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=−22x+5=0⇒x=−52x + 2 = 0 \Rightarrow x = -2\qquad 2x + 5 = 0 \Rightarrow x = -\frac{5}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−9±14x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{1}}{4}
  7. 7

    Simplify to get the solutions:

    x=−2orx=−52x = -2\quad\text{or}\quad x = -\frac{5}{2}
  8. 8

    Check by putting x = -2 back in: a·x² + b·x + c = 2·(4) + 9·(-2) + 10 = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 9x + 10 = 0\ \Rightarrow\ x = -2,\ -\frac{5}{2} \]
\[ 2x^2 + 9x + 10 = 0 \]
\[ D = 9^2 - 4\cdot2\cdot10 = 81 - 80 = 1 \]
\[ \left(x + 2\right)\left(2x + 5\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2\qquad 2x + 5 = 0 \Rightarrow x = -\frac{5}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{1}}{4} \]
\[ x = -2\quad\text{or}\quad x = -\frac{5}{2} \]

The parabola y = 2x² + 9x + 10

xy
-5.2517.875
-4.257.875
-3.251.875
-2.25-0.125
-1.251.875
-0.257.875
0.7517.875

Questions

What are the solutions of 2x² + 9x + 10 = 0?

The solutions of 2x² + 9x + 10 = 0 are x = -2 or x = -5/2.

Does 2x² + 9x + 10 = 0 factor?

Yes: (x+2)(2x+5) = 0.

What is the vertex of y = 2x² + 9x + 10?

(-9/4, -1/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -9/2, product = c/a = 5.

Similar equations

    Solve 2x² + 9x + 10 = 0: x = -2 or x = -5/2 (steps) | WordGPT