Alright! To solve the equation [tex]\(3 - 4x = 5x^2 - 14x\)[/tex] by completing the square, let's go through the steps.
1. Move all terms to one side of the equation to set it equal to zero:
[tex]\(3 - 4x - 5x^2 + 14x = 0\)[/tex]
Combine like terms:
[tex]\(0 = 5x^2 - 10x - 3\)[/tex]
So the correct step in the process is [tex]\(0 = 5x^2 - 10x - 3\)[/tex].
2. Rewrite the equation:
You can equivalently express it as:
[tex]\(3 - 4 = 5x^2 - 10x - 3\)[/tex]
3. Complete the square:
To make a perfect square trinomial, we take [tex]\(5(x^2 - 2x)\)[/tex]:
Add and subtract [tex]\(1\)[/tex] inside the parentheses:
[tex]\(5(x^2 - 2x + 1 - 1) - 3\)[/tex]
This transforms into:
[tex]\(5((x - 1)^2 - 1) - 3\)[/tex]
4. Distribute and Simplify:
Distribute 5:
[tex]\( 5(x - 1)^2 - 5 - 3\)[/tex]
5. Move the constant terms to the other side:
Move the constant terms to the other side to isolate the square term:
[tex]\(5(x - 1)^2 - 8 = 0\)[/tex] or
[tex]\(5(x - 1)^2 = 8\)[/tex]
6. Solve for [tex]\((x-1)^2\)[/tex]:
Divide by 5:
[tex]\(\frac{8}{5} = (x - 1)^2\)[/tex]
With all these steps in mind, we match them with the given options:
- [tex]\(3=5\left(x^2+2 x\right)\)[/tex] — Incorrect. This does not match our transformation.
- [tex]\(3=5 x^2-10 x\)[/tex] — Correct. It correctly represents the transformed equation.
- [tex]\(4=5\left(x^2-2 x+1\right)\)[/tex] — Incorrect. The constant on the left side is incorrect.
- [tex]\(8=5\left(x^2-2 x+1\right)\)[/tex] — Correct. It accurately represents after completing the square, and then moving constants.
- [tex]\(3=5(x-1)^2\)[/tex] — Incorrect. The constant on the left side is incorrect.
- [tex]\(4=5(x-1)^2\)[/tex] — Incorrect. The constant on the left side is incorrect.
- [tex]\(\frac{8}{5}=(x-1)^2\)[/tex] — Correct. It accurately shows the final step after dividing by 5.
So, the appropriate steps involved in the process are:
- [tex]\(3=5 x^2-10 x\)[/tex]
- [tex]\(8=5(x^2-2 x+1)\)[/tex]
- [tex]\(\frac{8}{5} = (x-1)^2\)[/tex]
These steps are correctly matched with the answer provided.
