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Solve -x² + 10x - 4 = 0

The solutions of -x² + 10x - 4 = 0 are x = 5 + √21 or x = 5 - √21, that is x ≈ 0.41742431 and x ≈ 9.5825757.

x² +x += 0
−x2+10x−4=0⇒x=5+21, 5−21-x^2 + 10x - 4 = 0 \Rightarrow x = 5 + \sqrt{21},\ 5 - \sqrt{21}
Discriminant D = 84 · x ≈ 0.417424, 9.58258

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

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Graph of y = -x² + 10x - 4-224681012-30-20-101020x = 0.41742431x = 9.5825757vertexy = -x² + 10x - 4
Graph of y = -x² + 10x - 4. The curve crosses the x-axis at x = 0.41742431 and x = 9.5825757; the vertex is at (5, 21).

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 = -4.

    −x2+10x−4=0-x^2 + 10x - 4 = 0
  2. 2

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

    D=102−4⋅(−1)⋅(−4)=100−16=84D = 10^2 - 4\cdot(-1)\cdot(-4) = 100 - 16 = 84
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−10±84−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{84}}{-2}
  5. 5

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

    x=5+21orx=5−21x = 5 + \sqrt{21}\quad\text{or}\quad x = 5 - \sqrt{21}
  6. 6

    As decimals: x ≈ 0.41742431 and x ≈ 9.5825757.

  7. 7

    Check numerically: for x ≈ 0.41742431 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 10x - 4 = 0\ \Rightarrow\ x = 5 + \sqrt{21},\ 5 - \sqrt{21} \]
\[ -x^2 + 10x - 4 = 0 \]
\[ D = 10^2 - 4\cdot(-1)\cdot(-4) = 100 - 16 = 84 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{84}}{-2} \]
\[ x = 5 + \sqrt{21}\quad\text{or}\quad x = 5 - \sqrt{21} \]

The parabola y = -x² + 10x - 4

xy
212
317
420
521
620
717
812

Questions

What are the solutions of -x² + 10x - 4 = 0?

The solutions of -x² + 10x - 4 = 0 are x = 5 + √21 or x = 5 - √21, that is x ≈ 0.41742431 and x ≈ 9.5825757.

Does -x² + 10x - 4 = 0 factor?

Not over the integers, because the discriminant 84 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 10x - 4?

(5, 21). 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 = 4.

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    Solve -x² + 10x - 4 = 0: x = 5 + √21 or x = 5 - √21 (steps) | WordGPT