WordGPT

Solve 2x² - 9x + 9 = 0

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

x² +x += 0
2x2−9x+9=0⇒x=3, 322x^2 - 9x + 9 = 0 \Rightarrow x = 3,\ \frac{3}{2}
Discriminant D = 9 · x ≈ 3, 1.5

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 + 9-1123455101520x = 3x = 1.5vertexy = 2x² - 9x + 9
Graph of y = 2x² - 9x + 9. The curve crosses the x-axis at x = 3 and x = 1.5; 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=02x^2 - 9x + 9 = 0
  2. 2

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

    D=(−9)2−4⋅2⋅9=81−72=9D = (-9)^2 - 4\cdot2\cdot9 = 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:

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

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=9±94x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{9}}{4}
  7. 7

    Simplify to get the solutions:

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

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

LaTeX source of the formulas on this page
\[ 2x^2 - 9x + 9 = 0\ \Rightarrow\ x = 3,\ \frac{3}{2} \]
\[ 2x^2 - 9x + 9 = 0 \]
\[ D = (-9)^2 - 4\cdot2\cdot9 = 81 - 72 = 9 \]
\[ \left(x - 3\right)\left(2x - 3\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad 2x - 3 = 0 \Rightarrow x = \frac{3}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{9}}{4} \]
\[ x = 3\quad\text{or}\quad x = \frac{3}{2} \]

The parabola y = 2x² - 9x + 9

xy
-0.7516.875
0.256.875
1.250.875
2.25-1.125
3.250.875
4.256.875
5.2516.875

Questions

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

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

Does 2x² - 9x + 9 = 0 factor?

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

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

(9/4, -9/8). The parabola opens upwards, so this is the minimum.

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 or x = 3/2 (steps) | WordGPT