WordGPT

Solve -2x² + 9x - 9 = 0

The solutions of -2x² + 9x - 9 = 0 are x = 3/2 or x = 3.

x² +x += 0
−2x2+9x−9=0⇒x=32, 3-2x^2 + 9x - 9 = 0 \Rightarrow x = \frac{3}{2},\ 3
Discriminant D = 9 · x ≈ 1.5, 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 = -2x² + 9x - 9-112345-20-15-10-5x = 1.5x = 3vertexy = -2x² + 9x - 9
Graph of y = -2x² + 9x - 9. The curve crosses the x-axis at x = 1.5 and x = 3; the vertex is at (2.25, 1.125).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -2, b = 9, c = -9.

    −2x2+9x−9=0-2x^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⋅(−2)⋅(−9)=81−72=9D = 9^2 - 4\cdot(-2)\cdot(-9) = 81 - 72 = 9
  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:

    −(2x−3)(x−3)=0-\left(2x - 3\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:

    2x−3=0⇒x=32x−3=0⇒x=32x - 3 = 0 \Rightarrow x = \frac{3}{2}\qquad x - 3 = 0 \Rightarrow x = 3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−9±9−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{9}}{-4}
  7. 7

    Simplify to get the solutions:

    x=32orx=3x = \frac{3}{2}\quad\text{or}\quad x = 3
  8. 8

    Check by putting x = 3/2 back in: a·x² + b·x + c = -2·(9/4) + 9·(3/2) + (-9) = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 9x - 9 = 0\ \Rightarrow\ x = \frac{3}{2},\ 3 \]
\[ -2x^2 + 9x - 9 = 0 \]
\[ D = 9^2 - 4\cdot(-2)\cdot(-9) = 81 - 72 = 9 \]
\[ -\left(2x - 3\right)\left(x - 3\right) = 0 \]
\[ 2x - 3 = 0 \Rightarrow x = \frac{3}{2}\qquad x - 3 = 0 \Rightarrow x = 3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{9}}{-4} \]
\[ x = \frac{3}{2}\quad\text{or}\quad x = 3 \]

The parabola y = -2x² + 9x - 9

xy
-0.75-16.875
0.25-6.875
1.25-0.875
2.251.125
3.25-0.875
4.25-6.875
5.25-16.875

Questions

What are the solutions of -2x² + 9x - 9 = 0?

The solutions of -2x² + 9x - 9 = 0 are x = 3/2 or x = 3.

Does -2x² + 9x - 9 = 0 factor?

Yes: -1(2x-3)(x-3) = 0.

What is the vertex of y = -2x² + 9x - 9?

(9/4, 9/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 9/2, product = c/a = 9/2.

Similar equations

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