WordGPT

Solve -2x² - 10x - 12 = 0

The solutions of -2x² - 10x - 12 = 0 are x = -3 or x = -2.

x² +x += 0
−2x2−10x−12=0⇒x=−3, −2-2x^2 - 10x - 12 = 0 \Rightarrow x = -3,\ -2
Discriminant D = 4 · x ≈ -3, -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 - 10x - 12-5-4-3-2-1-15-10-5x = -3x = -2vertexy = -2x² - 10x - 12
Graph of y = -2x² - 10x - 12. The curve crosses the x-axis at x = -3 and x = -2; the vertex is at (-2.5, 0.5).

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 = -10, c = -12.

    −2x2−10x−12=0-2x^2 - 10x - 12 = 0
  2. 2

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

    D=(−10)2−4⋅(−2)⋅(−12)=100−96=4D = (-10)^2 - 4\cdot(-2)\cdot(-12) = 100 - 96 = 4
  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:

    −2(x+3)(x+2)=0-2\left(x + 3\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:

    x+3=0⇒x=−3x+2=0⇒x=−2x + 3 = 0 \Rightarrow x = -3\qquad x + 2 = 0 \Rightarrow x = -2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=10±4−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{4}}{-4}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -3 back in: a·x² + b·x + c = -2·(9) + (-10)·(-3) + (-12) = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 10x - 12 = 0\ \Rightarrow\ x = -3,\ -2 \]
\[ -2x^2 - 10x - 12 = 0 \]
\[ D = (-10)^2 - 4\cdot(-2)\cdot(-12) = 100 - 96 = 4 \]
\[ -2\left(x + 3\right)\left(x + 2\right) = 0 \]
\[ x + 3 = 0 \Rightarrow x = -3\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{4}}{-4} \]
\[ x = -3\quad\text{or}\quad x = -2 \]

The parabola y = -2x² - 10x - 12

xy
-5.5-17.5
-4.5-7.5
-3.5-1.5
-2.50.5
-1.5-1.5
-0.5-7.5
0.5-17.5

Questions

What are the solutions of -2x² - 10x - 12 = 0?

The solutions of -2x² - 10x - 12 = 0 are x = -3 or x = -2.

Does -2x² - 10x - 12 = 0 factor?

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

What is the vertex of y = -2x² - 10x - 12?

(-5/2, 1/2). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -5, product = c/a = 6.

Similar equations

    Solve -2x² - 10x - 12 = 0: x = -3 or x = -2 (steps) | WordGPT