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Solve -x² + 8x - 4 = 0

The solutions of -x² + 8x - 4 = 0 are x = 4 + 2√3 or x = 4 - 2√3, that is x ≈ 0.53589838 and x ≈ 7.4641016.

x² +x += 0
−x2+8x−4=0⇒x=4+23, 4−23-x^2 + 8x - 4 = 0 \Rightarrow x = 4 + 2\sqrt{3},\ 4 - 2\sqrt{3}
Discriminant D = 48 · x ≈ 0.535898, 7.4641

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

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Graph of y = -x² + 8x - 42468-25-20-15-10-551015x = 0.53589838x = 7.4641016vertexy = -x² + 8x - 4
Graph of y = -x² + 8x - 4. The curve crosses the x-axis at x = 0.53589838 and x = 7.4641016; the vertex is at (4, 12).

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 = -4.

    −x2+8x−4=0-x^2 + 8x - 4 = 0
  2. 2

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

    D=82−4⋅(−1)⋅(−4)=64−16=48D = 8^2 - 4\cdot(-1)\cdot(-4) = 64 - 16 = 48
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−8±48−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{48}}{-2}
  5. 5

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

    x=4+23orx=4−23x = 4 + 2\sqrt{3}\quad\text{or}\quad x = 4 - 2\sqrt{3}
  6. 6

    As decimals: x ≈ 0.53589838 and x ≈ 7.4641016.

  7. 7

    Check numerically: for x ≈ 0.53589838 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 8x - 4 = 0\ \Rightarrow\ x = 4 + 2\sqrt{3},\ 4 - 2\sqrt{3} \]
\[ -x^2 + 8x - 4 = 0 \]
\[ D = 8^2 - 4\cdot(-1)\cdot(-4) = 64 - 16 = 48 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{48}}{-2} \]
\[ x = 4 + 2\sqrt{3}\quad\text{or}\quad x = 4 - 2\sqrt{3} \]

The parabola y = -x² + 8x - 4

xy
13
28
311
412
511
68
73

Questions

What are the solutions of -x² + 8x - 4 = 0?

The solutions of -x² + 8x - 4 = 0 are x = 4 + 2√3 or x = 4 - 2√3, that is x ≈ 0.53589838 and x ≈ 7.4641016.

Does -x² + 8x - 4 = 0 factor?

Not over the integers, because the discriminant 48 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 8x - 4?

(4, 12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 8, product = c/a = 4.

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    Solve -x² + 8x - 4 = 0: x = 4 + 2√3 or x = 4 - 2√3 (steps) | WordGPT