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Solve -x² + 5x + 6 = 0

The solutions of -x² + 5x + 6 = 0 are x = -1 or x = 6.

x² +x += 0
−x2+5x+6=0⇒x=−1, 6-x^2 + 5x + 6 = 0 \Rightarrow x = -1,\ 6
Discriminant D = 49 · x ≈ -1, 6

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

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Graph of y = -x² + 5x + 6-22468-25-20-15-10-551015x = -1x = 6vertexy = -x² + 5x + 6
Graph of y = -x² + 5x + 6. The curve crosses the x-axis at x = -1 and x = 6; the vertex is at (2.5, 12.25).

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 = 5, c = 6.

    −x2+5x+6=0-x^2 + 5x + 6 = 0
  2. 2

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

    D=52−4⋅(−1)⋅6=25+24=49D = 5^2 - 4\cdot(-1)\cdot6 = 25 + 24 = 49
  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+1)(x−6)=0-\left(x + 1\right)\left(x - 6\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+1=0⇒x=−1x−6=0⇒x=6x + 1 = 0 \Rightarrow x = -1\qquad x - 6 = 0 \Rightarrow x = 6
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−5±49−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{49}}{-2}
  7. 7

    Simplify to get the solutions:

    x=−1orx=6x = -1\quad\text{or}\quad x = 6
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = -1·(1) + 5·(-1) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 + 5x + 6 = 0\ \Rightarrow\ x = -1,\ 6 \]
\[ -x^2 + 5x + 6 = 0 \]
\[ D = 5^2 - 4\cdot(-1)\cdot6 = 25 + 24 = 49 \]
\[ -\left(x + 1\right)\left(x - 6\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x - 6 = 0 \Rightarrow x = 6 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-5 \pm \sqrt{49}}{-2} \]
\[ x = -1\quad\text{or}\quad x = 6 \]

The parabola y = -x² + 5x + 6

xy
-0.53.25
0.58.25
1.511.25
2.512.25
3.511.25
4.58.25
5.53.25

Questions

What are the solutions of -x² + 5x + 6 = 0?

The solutions of -x² + 5x + 6 = 0 are x = -1 or x = 6.

Does -x² + 5x + 6 = 0 factor?

Yes: -1(x+1)(x-6) = 0.

What is the vertex of y = -x² + 5x + 6?

(5/2, 49/4). 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.

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