WordGPT

Solve -3x² + 6x + 9 = 0

The solutions of -3x² + 6x + 9 = 0 are x = -1 or x = 3.

x² +x += 0
−3x2+6x+9=0⇒x=−1, 3-3x^2 + 6x + 9 = 0 \Rightarrow x = -1,\ 3
Discriminant D = 144 · x ≈ -1, 3

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 + 6x + 9-3-2-112345-50-40-30-20-101020x = -1x = 3vertexy = -3x² + 6x + 9
Graph of y = -3x² + 6x + 9. The curve crosses the x-axis at x = -1 and x = 3; the vertex is at (1, 12).

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 = 6, c = 9.

    −3x2+6x+9=0-3x^2 + 6x + 9 = 0
  2. 2

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

    D=62−4⋅(−3)⋅9=36+108=144D = 6^2 - 4\cdot(-3)\cdot9 = 36 + 108 = 144
  3. 3

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

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −3(x+1)(x−3)=0-3\left(x + 1\right)\left(x - 3\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x+1=0⇒x=−1x−3=0⇒x=3x + 1 = 0 \Rightarrow x = -1\qquad x - 3 = 0 \Rightarrow x = 3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−6±144−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{144}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−1orx=3x = -1\quad\text{or}\quad x = 3
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = -3·(1) + 6·(-1) + 9 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 6x + 9 = 0\ \Rightarrow\ x = -1,\ 3 \]
\[ -3x^2 + 6x + 9 = 0 \]
\[ D = 6^2 - 4\cdot(-3)\cdot9 = 36 + 108 = 144 \]
\[ -3\left(x + 1\right)\left(x - 3\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad x - 3 = 0 \Rightarrow x = 3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{144}}{-6} \]
\[ x = -1\quad\text{or}\quad x = 3 \]

The parabola y = -3x² + 6x + 9

xy
-2-15
-10
09
112
29
30
4-15

Questions

What are the solutions of -3x² + 6x + 9 = 0?

The solutions of -3x² + 6x + 9 = 0 are x = -1 or x = 3.

Does -3x² + 6x + 9 = 0 factor?

Yes: -3(x+1)(x-3) = 0.

What is the vertex of y = -3x² + 6x + 9?

(1, 12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 2, product = c/a = -3.

Similar equations

    Solve -3x² + 6x + 9 = 0: x = -1 or x = 3 (steps) | WordGPT