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Solve x² + 8x - 9 = 0

The solutions of x² + 8x - 9 = 0 are x = 1 or x = -9.

x² +x += 0
x2+8x−9=0⇒x=1, −9x^2 + 8x - 9 = 0 \Rightarrow x = 1,\ -9
Discriminant D = 100 · 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 + 8x - 9-10-8-6-4-22-30-20-10102030x = 1x = -9vertexy = x² + 8x - 9
Graph of y = x² + 8x - 9. The curve crosses the x-axis at x = 1 and x = -9; the vertex is at (-4, -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 = 8, c = -9.

    x2+8x−9=0x^2 + 8x - 9 = 0
  2. 2

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

    D=82−4⋅1⋅(−9)=64+36=100D = 8^2 - 4\cdot1\cdot(-9) = 64 + 36 = 100
  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=−8±1002x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{100}}{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) + 8·(1) + (-9) = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 + 8x - 9 = 0\ \Rightarrow\ x = 1,\ -9 \]
\[ x^2 + 8x - 9 = 0 \]
\[ D = 8^2 - 4\cdot1\cdot(-9) = 64 + 36 = 100 \]
\[ \left(x - 1\right)\left(x + 9\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad x + 9 = 0 \Rightarrow x = -9 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{100}}{2} \]
\[ x = 1\quad\text{or}\quad x = -9 \]

The parabola y = x² + 8x - 9

xy
-7-16
-6-21
-5-24
-4-25
-3-24
-2-21
-1-16

Questions

What are the solutions of x² + 8x - 9 = 0?

The solutions of x² + 8x - 9 = 0 are x = 1 or x = -9.

Does x² + 8x - 9 = 0 factor?

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

What is the vertex of y = x² + 8x - 9?

(-4, -25). 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 = -9.

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