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Solve 5x² - 6x - 10 = 0

The solutions of 5x² - 6x - 10 = 0 are x = (3 + √59)/5 or x = (3 - √59)/5, that is x ≈ 2.1362291 and x ≈ -0.93622915.

x² +x += 0
5x2−6x−10=0⇒x=3+595, 3−5955x^2 - 6x - 10 = 0 \Rightarrow x = \frac{3 + \sqrt{59}}{5},\ \frac{3 - \sqrt{59}}{5}
Discriminant D = 236 · x ≈ 2.13623, -0.936229

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

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Graph of y = 5x² - 6x - 10-3-2-11234-20204060x = 2.1362291x = -0.93622915vertexy = 5x² - 6x - 10
Graph of y = 5x² - 6x - 10. The curve crosses the x-axis at x = 2.1362291 and x = -0.93622915; the vertex is at (0.6, -11.8).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = -6, c = -10.

    5x2−6x−10=05x^2 - 6x - 10 = 0
  2. 2

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

    D=(−6)2−4⋅5⋅(−10)=36+200=236D = (-6)^2 - 4\cdot5\cdot(-10) = 36 + 200 = 236
  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±23610x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{236}}{10}
  5. 5

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

    x=3+595orx=3−595x = \frac{3 + \sqrt{59}}{5}\quad\text{or}\quad x = \frac{3 - \sqrt{59}}{5}
  6. 6

    As decimals: x ≈ 2.1362291 and x ≈ -0.93622915.

  7. 7

    Check numerically: for x ≈ 2.1362291 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 6x - 10 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{59}}{5},\ \frac{3 - \sqrt{59}}{5} \]
\[ 5x^2 - 6x - 10 = 0 \]
\[ D = (-6)^2 - 4\cdot5\cdot(-10) = 36 + 200 = 236 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{236}}{10} \]
\[ x = \frac{3 + \sqrt{59}}{5}\quad\text{or}\quad x = \frac{3 - \sqrt{59}}{5} \]

The parabola y = 5x² - 6x - 10

xy
-2.433.2
-1.48.2
-0.4-6.8
0.6-11.8
1.6-6.8
2.68.2
3.633.2

Questions

What are the solutions of 5x² - 6x - 10 = 0?

The solutions of 5x² - 6x - 10 = 0 are x = (3 + √59)/5 or x = (3 - √59)/5, that is x ≈ 2.1362291 and x ≈ -0.93622915.

Does 5x² - 6x - 10 = 0 factor?

Not over the integers, because the discriminant 236 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 5x² - 6x - 10?

(3/5, -59/5). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 6/5, product = c/a = -2.

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    Solve 5x² - 6x - 10 = 0: x = (3 + √59)/5 or x = (3 - √59)/5 (steps) | WordGPT