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Solve x² + 10x + 9 = 0

The solutions of x² + 10x + 9 = 0 are x = -1 or x = -9.

x² +x += 0
x2+10x+9=0⇒x=−1, −9x^2 + 10x + 9 = 0 \Rightarrow x = -1,\ -9
Discriminant D = 64 · x ≈ -1, -9

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

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Graph of y = x² + 10x + 9-10-8-6-4-2-20-101020x = -1x = -9vertexy = x² + 10x + 9
Graph of y = x² + 10x + 9. The curve crosses the x-axis at x = -1 and x = -9; the vertex is at (-5, -16).

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 = 10, c = 9.

    x2+10x+9=0x^2 + 10x + 9 = 0
  2. 2

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

    D=102−4⋅1⋅9=100−36=64D = 10^2 - 4\cdot1\cdot9 = 100 - 36 = 64
  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+9)=0\left(x + 1\right)\left(x + 9\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+9=0⇒x=−9x + 1 = 0 \Rightarrow x = -1\qquad x + 9 = 0 \Rightarrow x = -9
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−10±642x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{64}}{2}
  7. 7

    Simplify to get the solutions:

    x=−1orx=−9x = -1\quad\text{or}\quad x = -9
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = 1·(1) + 10·(-1) + 9 = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 + 10x + 9 = 0\ \Rightarrow\ x = -1,\ -9 \]
\[ x^2 + 10x + 9 = 0 \]
\[ D = 10^2 - 4\cdot1\cdot9 = 100 - 36 = 64 \]
\[ \left(x + 1\right)\left(x + 9\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x + 9 = 0 \Rightarrow x = -9 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{64}}{2} \]
\[ x = -1\quad\text{or}\quad x = -9 \]

The parabola y = x² + 10x + 9

xy
-8-7
-7-12
-6-15
-5-16
-4-15
-3-12
-2-7

Questions

What are the solutions of x² + 10x + 9 = 0?

The solutions of x² + 10x + 9 = 0 are x = -1 or x = -9.

Does x² + 10x + 9 = 0 factor?

Yes: (x+1)(x+9) = 0.

What is the vertex of y = x² + 10x + 9?

(-5, -16). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -10, product = c/a = 9.

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