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Solve -x² + 12x - 5 = 0

The solutions of -x² + 12x - 5 = 0 are x = 6 + √31 or x = 6 - √31, that is x ≈ 0.43223564 and x ≈ 11.567764.

x² +x += 0
−x2+12x−5=0⇒x=6+31, 6−31-x^2 + 12x - 5 = 0 \Rightarrow x = 6 + \sqrt{31},\ 6 - \sqrt{31}
Discriminant D = 124 · x ≈ 0.432236, 11.5678

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

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Graph of y = -x² + 12x - 5-22468101214-30-20-10102030x = 0.43223564x = 11.567764vertexy = -x² + 12x - 5
Graph of y = -x² + 12x - 5. The curve crosses the x-axis at x = 0.43223564 and x = 11.567764; the vertex is at (6, 31).

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 = 12, c = -5.

    −x2+12x−5=0-x^2 + 12x - 5 = 0
  2. 2

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

    D=122−4⋅(−1)⋅(−5)=144−20=124D = 12^2 - 4\cdot(-1)\cdot(-5) = 144 - 20 = 124
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−12±124−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{124}}{-2}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=6+31orx=6−31x = 6 + \sqrt{31}\quad\text{or}\quad x = 6 - \sqrt{31}
  6. 6

    As decimals: x ≈ 0.43223564 and x ≈ 11.567764.

  7. 7

    Check numerically: for x ≈ 0.43223564 the left side is 4.4408921e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 12x - 5 = 0\ \Rightarrow\ x = 6 + \sqrt{31},\ 6 - \sqrt{31} \]
\[ -x^2 + 12x - 5 = 0 \]
\[ D = 12^2 - 4\cdot(-1)\cdot(-5) = 144 - 20 = 124 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{124}}{-2} \]
\[ x = 6 + \sqrt{31}\quad\text{or}\quad x = 6 - \sqrt{31} \]

The parabola y = -x² + 12x - 5

xy
322
427
530
631
730
827
922

Questions

What are the solutions of -x² + 12x - 5 = 0?

The solutions of -x² + 12x - 5 = 0 are x = 6 + √31 or x = 6 - √31, that is x ≈ 0.43223564 and x ≈ 11.567764.

Does -x² + 12x - 5 = 0 factor?

Not over the integers, because the discriminant 124 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 12x - 5?

(6, 31). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 12, product = c/a = 5.

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    Solve -x² + 12x - 5 = 0: x = 6 + √31 or x = 6 - √31 (steps) | WordGPT