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Solve 3x² - 10x - 9 = 0

The solutions of 3x² - 10x - 9 = 0 are x = (5 + 2√13)/3 or x = (5 - 2√13)/3, that is x ≈ 4.0703675 and x ≈ -0.73703418.

x² +x += 0
3x2−10x−9=0⇒x=5+2133, 5−21333x^2 - 10x - 9 = 0 \Rightarrow x = \frac{5 + 2\sqrt{13}}{3},\ \frac{5 - 2\sqrt{13}}{3}
Discriminant D = 208 · x ≈ 4.07037, -0.737034

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

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Graph of y = 3x² - 10x - 9-3-2-1123456-20-10102030405060x = 4.0703675x = -0.73703418vertexy = 3x² - 10x - 9
Graph of y = 3x² - 10x - 9. The curve crosses the x-axis at x = 4.0703675 and x = -0.73703418; the vertex is at (1.6666667, -17.333333).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 3, b = -10, c = -9.

    3x2−10x−9=03x^2 - 10x - 9 = 0
  2. 2

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

    D=(−10)2−4⋅3⋅(−9)=100+108=208D = (-10)^2 - 4\cdot3\cdot(-9) = 100 + 108 = 208
  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±2086x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{208}}{6}
  5. 5

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

    x=5+2133orx=5−2133x = \frac{5 + 2\sqrt{13}}{3}\quad\text{or}\quad x = \frac{5 - 2\sqrt{13}}{3}
  6. 6

    As decimals: x ≈ 4.0703675 and x ≈ -0.73703418.

  7. 7

    Check numerically: for x ≈ 4.0703675 the left side is 7.1054274e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 10x - 9 = 0\ \Rightarrow\ x = \frac{5 + 2\sqrt{13}}{3},\ \frac{5 - 2\sqrt{13}}{3} \]
\[ 3x^2 - 10x - 9 = 0 \]
\[ D = (-10)^2 - 4\cdot3\cdot(-9) = 100 + 108 = 208 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{208}}{6} \]
\[ x = \frac{5 + 2\sqrt{13}}{3}\quad\text{or}\quad x = \frac{5 - 2\sqrt{13}}{3} \]

The parabola y = 3x² - 10x - 9

xy
-1.33339.6660667
-0.3333-5.3337333
0.6667-14.333533
1.6667-17.333333
2.6667-14.333133
3.6667-5.3329333
4.66679.6672667

Questions

What are the solutions of 3x² - 10x - 9 = 0?

The solutions of 3x² - 10x - 9 = 0 are x = (5 + 2√13)/3 or x = (5 - 2√13)/3, that is x ≈ 4.0703675 and x ≈ -0.73703418.

Does 3x² - 10x - 9 = 0 factor?

Not over the integers, because the discriminant 208 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 3x² - 10x - 9?

(5/3, -52/3). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 10/3, product = c/a = -3.

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    Solve 3x² - 10x - 9 = 0: x = (5 + 2√13)/3 or x = (5 - 2√13)/3 (steps) | WordGPT