WordGPT

Solve 3x² + 9x - 9 = 0

The solutions of 3x² + 9x - 9 = 0 are x = (-3 + √21)/2 or x = (-3 - √21)/2, that is x ≈ 0.79128785 and x ≈ -3.7912878.

x² +x += 0
3x2+9x−9=0⇒x=−3+212, −3−2123x^2 + 9x - 9 = 0 \Rightarrow x = \frac{-3 + \sqrt{21}}{2},\ \frac{-3 - \sqrt{21}}{2}
Discriminant D = 189 · x ≈ 0.791288, -3.79129

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

Explain it step by step in chatMake a short video of it
Graph of y = 3x² + 9x - 9-6-5-4-3-2-1123-20-101020304050x = 0.79128785x = -3.7912878vertexy = 3x² + 9x - 9
Graph of y = 3x² + 9x - 9. The curve crosses the x-axis at x = 0.79128785 and x = -3.7912878; the vertex is at (-1.5, -15.75).

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 = 9, c = -9.

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

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

    D=92−4⋅3⋅(−9)=81+108=189D = 9^2 - 4\cdot3\cdot(-9) = 81 + 108 = 189
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−9±1896x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{189}}{6}
  5. 5

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

    x=−3+212orx=−3−212x = \frac{-3 + \sqrt{21}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{21}}{2}
  6. 6

    As decimals: x ≈ 0.79128785 and x ≈ -3.7912878.

  7. 7

    Check numerically: for x ≈ 0.79128785 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 3x^2 + 9x - 9 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{21}}{2},\ \frac{-3 - \sqrt{21}}{2} \]
\[ 3x^2 + 9x - 9 = 0 \]
\[ D = 9^2 - 4\cdot3\cdot(-9) = 81 + 108 = 189 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{189}}{6} \]
\[ x = \frac{-3 + \sqrt{21}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{21}}{2} \]

The parabola y = 3x² + 9x - 9

xy
-4.511.25
-3.5-3.75
-2.5-12.75
-1.5-15.75
-0.5-12.75
0.5-3.75
1.511.25

Questions

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

The solutions of 3x² + 9x - 9 = 0 are x = (-3 + √21)/2 or x = (-3 - √21)/2, that is x ≈ 0.79128785 and x ≈ -3.7912878.

Does 3x² + 9x - 9 = 0 factor?

Not over the integers, because the discriminant 189 is not a perfect square. Use the quadratic formula.

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

(-3/2, -63/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = -3.

Similar equations

    Solve 3x² + 9x - 9 = 0: x = (-3 + √21)/2 or x = (-3 - √21)/2 (steps) | WordGPT