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Solve x² + 8x + 7 = 0

The solutions of x² + 8x + 7 = 0 are x = -1 or x = -7.

x² +x += 0
x2+8x+7=0⇒x=−1, −7x^2 + 8x + 7 = 0 \Rightarrow x = -1,\ -7
Discriminant D = 36 · x ≈ -1, -7

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

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Graph of y = x² + 8x + 7-8-6-4-2-10-55101520x = -1x = -7vertexy = x² + 8x + 7
Graph of y = x² + 8x + 7. The curve crosses the x-axis at x = -1 and x = -7; the vertex is at (-4, -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 = 8, c = 7.

    x2+8x+7=0x^2 + 8x + 7 = 0
  2. 2

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

    D=82−4⋅1⋅7=64−28=36D = 8^2 - 4\cdot1\cdot7 = 64 - 28 = 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+1)(x+7)=0\left(x + 1\right)\left(x + 7\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+7=0⇒x=−7x + 1 = 0 \Rightarrow x = -1\qquad x + 7 = 0 \Rightarrow x = -7
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±362x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{36}}{2}
  7. 7

    Simplify to get the solutions:

    x=−1orx=−7x = -1\quad\text{or}\quad x = -7
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = 1·(1) + 8·(-1) + 7 = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 + 8x + 7 = 0\ \Rightarrow\ x = -1,\ -7 \]
\[ x^2 + 8x + 7 = 0 \]
\[ D = 8^2 - 4\cdot1\cdot7 = 64 - 28 = 36 \]
\[ \left(x + 1\right)\left(x + 7\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x + 7 = 0 \Rightarrow x = -7 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{36}}{2} \]
\[ x = -1\quad\text{or}\quad x = -7 \]

The parabola y = x² + 8x + 7

xy
-70
-6-5
-5-8
-4-9
-3-8
-2-5
-10

Questions

What are the solutions of x² + 8x + 7 = 0?

The solutions of x² + 8x + 7 = 0 are x = -1 or x = -7.

Does x² + 8x + 7 = 0 factor?

Yes: (x+1)(x+7) = 0.

What is the vertex of y = x² + 8x + 7?

(-4, -9). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -8, product = c/a = 7.

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