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Solve x² - 5x - 6 = 0

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

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

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

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Graph of y = x² - 5x - 6-22468-15-10-5510152025x = 6x = -1vertexy = x² - 5x - 6
Graph of y = x² - 5x - 6. The curve crosses the x-axis at x = 6 and x = -1; 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=0x^2 - 5x - 6 = 0
  2. 2

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

    D=(−5)2−4⋅1⋅(−6)=25+24=49D = (-5)^2 - 4\cdot1\cdot(-6) = 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−6)(x+1)=0\left(x - 6\right)\left(x + 1\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−6=0⇒x=6x+1=0⇒x=−1x - 6 = 0 \Rightarrow x = 6\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=5±492x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{49}}{2}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = 6 back in: a·x² + b·x + c = 1·(36) + (-5)·(6) + (-6) = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 - 5x - 6 = 0\ \Rightarrow\ x = 6,\ -1 \]
\[ x^2 - 5x - 6 = 0 \]
\[ D = (-5)^2 - 4\cdot1\cdot(-6) = 25 + 24 = 49 \]
\[ \left(x - 6\right)\left(x + 1\right) = 0 \]
\[ x - 6 = 0 \Rightarrow x = 6\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{49}}{2} \]
\[ x = 6\quad\text{or}\quad x = -1 \]

The parabola y = x² - 5x - 6

xy
-0.5-3.25
0.5-8.25
1.5-11.25
2.5-12.25
3.5-11.25
4.5-8.25
5.5-3.25

Questions

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

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

Does x² - 5x - 6 = 0 factor?

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

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

(5/2, -49/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 5, product = c/a = -6.

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