WordGPT

Solve -x² + x + 6 = 0

The solutions of -x² + x + 6 = 0 are x = -2 or x = 3.

x² +x += 0
−x2+x+6=0⇒x=−2, 3-x^2 + x + 6 = 0 \Rightarrow x = -2,\ 3
Discriminant D = 25 · x ≈ -2, 3

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 = -x² + x + 6-4-3-2-112345-20-15-10-55x = -2x = 3vertexy = -x² + x + 6
Graph of y = -x² + x + 6. The curve crosses the x-axis at x = -2 and x = 3; the vertex is at (0.5, 6.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = 1, c = 6.

    −x2+x+6=0-x^2 + x + 6 = 0
  2. 2

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

    D=12−4⋅(−1)⋅6=1+24=25D = 1^2 - 4\cdot(-1)\cdot6 = 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:

    −(x+2)(x−3)=0-\left(x + 2\right)\left(x - 3\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=−2x−3=0⇒x=3x + 2 = 0 \Rightarrow x = -2\qquad x - 3 = 0 \Rightarrow x = 3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

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

    Check by putting x = -2 back in: a·x² + b·x + c = -1·(4) + 1·(-2) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 + x + 6 = 0\ \Rightarrow\ x = -2,\ 3 \]
\[ -x^2 + x + 6 = 0 \]
\[ D = 1^2 - 4\cdot(-1)\cdot6 = 1 + 24 = 25 \]
\[ -\left(x + 2\right)\left(x - 3\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2\qquad x - 3 = 0 \Rightarrow x = 3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{25}}{-2} \]
\[ x = -2\quad\text{or}\quad x = 3 \]

The parabola y = -x² + x + 6

xy
-2.5-2.75
-1.52.25
-0.55.25
0.56.25
1.55.25
2.52.25
3.5-2.75

Questions

What are the solutions of -x² + x + 6 = 0?

The solutions of -x² + x + 6 = 0 are x = -2 or x = 3.

Does -x² + x + 6 = 0 factor?

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

What is the vertex of y = -x² + x + 6?

(1/2, 25/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = -6.

Similar equations

    Solve -x² + x + 6 = 0: x = -2 or x = 3 (steps) | WordGPT