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Solve -x² + 6x - 4 = 0

The solutions of -x² + 6x - 4 = 0 are x = 3 + √5 or x = 3 - √5, that is x ≈ 0.76393202 and x ≈ 5.236068.

x² +x += 0
−x2+6x−4=0⇒x=3+5, 3−5-x^2 + 6x - 4 = 0 \Rightarrow x = 3 + \sqrt{5},\ 3 - \sqrt{5}
Discriminant D = 20 · x ≈ 0.763932, 5.23607

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

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Graph of y = -x² + 6x - 4-11234567-15-10-55x = 0.76393202x = 5.236068vertexy = -x² + 6x - 4
Graph of y = -x² + 6x - 4. The curve crosses the x-axis at x = 0.76393202 and x = 5.236068; the vertex is at (3, 5).

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 = -4.

    −x2+6x−4=0-x^2 + 6x - 4 = 0
  2. 2

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

    D=62−4⋅(−1)⋅(−4)=36−16=20D = 6^2 - 4\cdot(-1)\cdot(-4) = 36 - 16 = 20
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−6±20−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{20}}{-2}
  5. 5

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

    x=3+5orx=3−5x = 3 + \sqrt{5}\quad\text{or}\quad x = 3 - \sqrt{5}
  6. 6

    As decimals: x ≈ 0.76393202 and x ≈ 5.236068.

  7. 7

    Check numerically: for x ≈ 0.76393202 the left side is -4.4408921e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 6x - 4 = 0\ \Rightarrow\ x = 3 + \sqrt{5},\ 3 - \sqrt{5} \]
\[ -x^2 + 6x - 4 = 0 \]
\[ D = 6^2 - 4\cdot(-1)\cdot(-4) = 36 - 16 = 20 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{20}}{-2} \]
\[ x = 3 + \sqrt{5}\quad\text{or}\quad x = 3 - \sqrt{5} \]

The parabola y = -x² + 6x - 4

xy
0-4
11
24
35
44
51
6-4

Questions

What are the solutions of -x² + 6x - 4 = 0?

The solutions of -x² + 6x - 4 = 0 are x = 3 + √5 or x = 3 - √5, that is x ≈ 0.76393202 and x ≈ 5.236068.

Does -x² + 6x - 4 = 0 factor?

Not over the integers, because the discriminant 20 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 6x - 4?

(3, 5). 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 = 4.

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    Solve -x² + 6x - 4 = 0: x = 3 + √5 or x = 3 - √5 (steps) | WordGPT