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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, −3x^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.

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Graph of y = x² + x - 6-5-4-3-2-11234-55101520x = 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=0x^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\cdot1\cdot(-6) = 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±252x = \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
\[ x^2 + x - 6 = 0\ \Rightarrow\ x = 2,\ -3 \]
\[ x^2 + x - 6 = 0 \]
\[ D = 1^2 - 4\cdot1\cdot(-6) = 1 + 24 = 25 \]
\[ \left(x - 2\right)\left(x + 3\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad x + 3 = 0 \Rightarrow x = -3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{25}}{2} \]
\[ x = 2\quad\text{or}\quad x = -3 \]

The parabola y = x² + x - 6

xy
-3.52.75
-2.5-2.25
-1.5-5.25
-0.5-6.25
0.5-5.25
1.5-2.25
2.52.75

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

What is the vertex of y = x² + x - 6?

(-1/2, -25/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -1, product = c/a = -6.

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