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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, 9-x^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 - 9246810-20-101020x = 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=0-x^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\cdot(-1)\cdot(-9) = 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±64−2x = \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
\[ -x^2 + 10x - 9 = 0\ \Rightarrow\ x = 1,\ 9 \]
\[ -x^2 + 10x - 9 = 0 \]
\[ D = 10^2 - 4\cdot(-1)\cdot(-9) = 100 - 36 = 64 \]
\[ -\left(x - 1\right)\left(x - 9\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad x - 9 = 0 \Rightarrow x = 9 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{64}}{-2} \]
\[ x = 1\quad\text{or}\quad x = 9 \]

The parabola y = -x² + 10x - 9

xy
27
312
415
516
615
712
87

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

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

(5, 16). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 10, product = c/a = 9.

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