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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 ≈ 0.73703418 and x ≈ -4.0703675.

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 ≈ 0.737034, -4.07037

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

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Graph of y = 3x² + 10x - 9-6-5-4-3-2-1123-20-10102030405060x = 0.73703418x = -4.0703675vertexy = 3x² + 10x - 9
Graph of y = 3x² + 10x - 9. The curve crosses the x-axis at x = 0.73703418 and x = -4.0703675; 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=102−4⋅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 ≈ 0.73703418 and x ≈ -4.0703675.

  7. 7

    Check numerically: for x ≈ 0.73703418 the left side is -1.7763568e-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
-4.66679.6672667
-3.6667-5.3329333
-2.6667-14.333133
-1.6667-17.333333
-0.6667-14.333533
0.3333-5.3337333
1.33339.6660667

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 ≈ 0.73703418 and x ≈ -4.0703675.

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