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Solve x² + 7x + 6 = 0

The solutions of x² + 7x + 6 = 0 are x = -1 or x = -6.

x² +x += 0
x2+7x+6=0⇒x=−1, −6x^2 + 7x + 6 = 0 \Rightarrow x = -1,\ -6
Discriminant D = 25 · 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 + 7x + 6-8-7-6-5-4-3-2-11-55101520x = -1x = -6vertexy = x² + 7x + 6
Graph of y = x² + 7x + 6. The curve crosses the x-axis at x = -1 and x = -6; the vertex is at (-3.5, -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 = 7, c = 6.

    x2+7x+6=0x^2 + 7x + 6 = 0
  2. 2

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

    D=72−4⋅1⋅6=49−24=25D = 7^2 - 4\cdot1\cdot6 = 49 - 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+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=−7±252x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{25}}{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) + 7·(-1) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 + 7x + 6 = 0\ \Rightarrow\ x = -1,\ -6 \]
\[ x^2 + 7x + 6 = 0 \]
\[ D = 7^2 - 4\cdot1\cdot6 = 49 - 24 = 25 \]
\[ \left(x + 1\right)\left(x + 6\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x + 6 = 0 \Rightarrow x = -6 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{25}}{2} \]
\[ x = -1\quad\text{or}\quad x = -6 \]

The parabola y = x² + 7x + 6

xy
-6.52.75
-5.5-2.25
-4.5-5.25
-3.5-6.25
-2.5-5.25
-1.5-2.25
-0.52.75

Questions

What are the solutions of x² + 7x + 6 = 0?

The solutions of x² + 7x + 6 = 0 are x = -1 or x = -6.

Does x² + 7x + 6 = 0 factor?

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

What is the vertex of y = x² + 7x + 6?

(-7/2, -25/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -7, product = c/a = 6.

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