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Solve x² + 6x = 0

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

x² +x += 0
x2+6x=0⇒x=0, −6x^2 + 6x = 0 \Rightarrow x = 0,\ -6
Discriminant D = 36 · x ≈ 0, -6

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

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Graph of y = x² + 6x-8-6-4-22-10-55101520x = 0x = -6vertexy = x² + 6x
Graph of y = x² + 6x. The curve crosses the x-axis at x = 0 and x = -6; the vertex is at (-3, -9).

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 = 6, c = 0.

    x2+6x=0x^2 + 6x = 0
  2. 2

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

    D=62−4⋅1⋅0=36+0=36D = 6^2 - 4\cdot1\cdot0 = 36 + 0 = 36
  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)(x+6)=0\left(x\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=0⇒x=0x+6=0⇒x=−6x = 0 \Rightarrow x = 0\qquad x + 6 = 0 \Rightarrow x = -6
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−6±362x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{36}}{2}
  7. 7

    Simplify to get the solutions:

    x=0orx=−6x = 0\quad\text{or}\quad x = -6
  8. 8

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

LaTeX source of the formulas on this page
\[ x^2 + 6x = 0\ \Rightarrow\ x = 0,\ -6 \]
\[ x^2 + 6x = 0 \]
\[ D = 6^2 - 4\cdot1\cdot0 = 36 + 0 = 36 \]
\[ \left(x\right)\left(x + 6\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad x + 6 = 0 \Rightarrow x = -6 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{36}}{2} \]
\[ x = 0\quad\text{or}\quad x = -6 \]

The parabola y = x² + 6x

xy
-60
-5-5
-4-8
-3-9
-2-8
-1-5
00

Questions

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

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

Does x² + 6x = 0 factor?

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

What is the vertex of y = x² + 6x?

(-3, -9). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -6, product = c/a = 0.

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