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Solve -x² + 6x - 9 = 0

The solution of -x² + 6x - 9 = 0 is x = 3 (a repeated root).

x² +x += 0
−x2+6x−9=0⇒x=3-x^2 + 6x - 9 = 0 \Rightarrow x = 3
Discriminant D = 0 · x ≈ 3

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

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Graph of y = -x² + 6x - 90.511.522.533.544.555.5-6-5-4-3-2-1x = 3vertexy = -x² + 6x - 9
Graph of y = -x² + 6x - 9. The curve crosses the x-axis at x = 3; the vertex is at (3, 0).

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 = -9.

    −x2+6x−9=0-x^2 + 6x - 9 = 0
  2. 2

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

    D=62−4⋅(−1)⋅(−9)=36−36=0D = 6^2 - 4\cdot(-1)\cdot(-9) = 36 - 36 = 0
  3. 3

    D is zero, so there is exactly one solution (a repeated root).

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −(x−3)(x−3)=0-\left(x - 3\right)\left(x - 3\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−3=0⇒x=3x - 3 = 0 \Rightarrow x = 3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−6±0−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{0}}{-2}
  7. 7

    Simplify to get the solution:

    x=3x = 3
  8. 8

    Check by putting x = 3 back in: a·x² + b·x + c = -1·(9) + 6·(3) + (-9) = 0 ✓

LaTeX source of the formulas on this page
\[ -x^2 + 6x - 9 = 0\ \Rightarrow\ x = 3 \]
\[ -x^2 + 6x - 9 = 0 \]
\[ D = 6^2 - 4\cdot(-1)\cdot(-9) = 36 - 36 = 0 \]
\[ -\left(x - 3\right)\left(x - 3\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{0}}{-2} \]
\[ x = 3 \]

The parabola y = -x² + 6x - 9

xy
0-9
1-4
2-1
30
4-1
5-4
6-9

Questions

What are the solutions of -x² + 6x - 9 = 0?

The solution of -x² + 6x - 9 = 0 is x = 3 (a repeated root).

Does -x² + 6x - 9 = 0 factor?

Yes: -1(x-3)(x-3) = 0.

What is the vertex of y = -x² + 6x - 9?

(3, 0). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 6, product = c/a = 9.

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