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Solve -x² + 10x - 3 = 0

The solutions of -x² + 10x - 3 = 0 are x = 5 + √22 or x = 5 - √22, that is x ≈ 0.30958424 and x ≈ 9.6904158.

x² +x += 0
−x2+10x−3=0⇒x=5+22, 5−22-x^2 + 10x - 3 = 0 \Rightarrow x = 5 + \sqrt{22},\ 5 - \sqrt{22}
Discriminant D = 88 · x ≈ 0.309584, 9.69042

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

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Graph of y = -x² + 10x - 3-224681012-30-20-101020x = 0.30958424x = 9.6904158vertexy = -x² + 10x - 3
Graph of y = -x² + 10x - 3. The curve crosses the x-axis at x = 0.30958424 and x = 9.6904158; the vertex is at (5, 22).

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 = -3.

    −x2+10x−3=0-x^2 + 10x - 3 = 0
  2. 2

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

    D=102−4⋅(−1)⋅(−3)=100−12=88D = 10^2 - 4\cdot(-1)\cdot(-3) = 100 - 12 = 88
  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±88−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{88}}{-2}
  5. 5

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

    x=5+22orx=5−22x = 5 + \sqrt{22}\quad\text{or}\quad x = 5 - \sqrt{22}
  6. 6

    As decimals: x ≈ 0.30958424 and x ≈ 9.6904158.

  7. 7

    Check numerically: for x ≈ 0.30958424 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -x^2 + 10x - 3 = 0\ \Rightarrow\ x = 5 + \sqrt{22},\ 5 - \sqrt{22} \]
\[ -x^2 + 10x - 3 = 0 \]
\[ D = 10^2 - 4\cdot(-1)\cdot(-3) = 100 - 12 = 88 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{88}}{-2} \]
\[ x = 5 + \sqrt{22}\quad\text{or}\quad x = 5 - \sqrt{22} \]

The parabola y = -x² + 10x - 3

xy
213
318
421
522
621
718
813

Questions

What are the solutions of -x² + 10x - 3 = 0?

The solutions of -x² + 10x - 3 = 0 are x = 5 + √22 or x = 5 - √22, that is x ≈ 0.30958424 and x ≈ 9.6904158.

Does -x² + 10x - 3 = 0 factor?

Not over the integers, because the discriminant 88 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 10x - 3?

(5, 22). 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 = 3.

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    Solve -x² + 10x - 3 = 0: x = 5 + √22 or x = 5 - √22 (steps) | WordGPT