y=3x^2=5x-1
how to sole it with partial factoring?
To solve the equation y = 3x^2 + 5x - 1 using partial factoring, we need to set the equation equal to zero and rearrange it in the form ax^2 + bx + c = 0.
1. Start with the original equation y = 3x^2 + 5x - 1.
2. Subtract y from both sides to set the equation equal to zero: 3x^2 + 5x - 1 - y = 0.
3. Group the terms together: 3x^2 + 5x - (1 + y) = 0.
4. Now, we need to find two numbers, let's say A and B, such that A * B = 3 and A + B = 5. These numbers will help us rewrite the middle term using partial factoring.
5. We have A = 3 and B = 1 (since 3 * 1 = 3 and 3 + 1 = 4).
6. Rewrite the middle term using the values of A and B: 3x^2 + 3x + 2x - (1 + y) = 0.
7. Now, we can factor by grouping. Take out the greatest common factor from the first two terms and the second two terms separately:
(3x^2 + 3x) + (2x - (1 + y)) = 0.
3x(x + 1) + (2x - (1 + y)) = 0.
8. Factor out a -1 from the 2x - (1 + y) term: 3x(x + 1) - (-1)(2x - (1 + y)) = 0.
9. Simplify: 3x(x + 1) + (1 + y)(2x - 1) = 0.
10. Now, we can set each factor equal to zero since the product of multiple terms is equal to zero if and only if at least one of the terms is zero:
3x = 0 or x + 1 = 0 or 2x - 1 = 0 or 1 + y = 0.
11. Solve each equation:
- For 3x = 0, divide both sides by 3 to get x = 0.
- For x + 1 = 0, subtract 1 from both sides to get x = -1.
- For 2x - 1 = 0, add 1 to both sides and then divide by 2 to get x = 1/2.
- For 1 + y = 0, subtract 1 from both sides to get y = -1.
So, the solutions to the equation y = 3x^2 + 5x - 1 using partial factoring are x = 0, x = -1, and x = 1/2.