how to solve 3x^2+4x-2=0 (mod 5)
To solve the equation 3x^2 + 4x - 2 ≡ 0 (mod 5), we need to find the values of x that satisfy the congruence relation modulo 5.
Step 1: Simplify the equation
In this step, we need to simplify the equation so that it becomes easier to work with.
3x^2 + 4x - 2 ≡ 0 (mod 5)
Step 2: Multiply the entire equation by the modular inverse of 3 modulo 5
To isolate the x terms, we can multiply the equation by the modular inverse of 3 modulo 5. The modular inverse of 3 (mod 5) is 2, as 3 * 2 ≡ 1 (mod 5).
(3x^2 + 4x - 2) * 2 ≡ 0 * 2 (mod 5)
6x^2 + 8x - 4 ≡ 0 (mod 5)
Step 3: Simplify the equation further
We can simplify the equation by reducing the coefficients modulo 5 and eliminating any unnecessary terms.
6x^2 + 8x - 4 ≡ x^2 + 3x - 4 ≡ 0 (mod 5)
Step 4: Factor or use the quadratic formula
Now that we have a simplified equation, we can either try to factor it or use the quadratic formula to solve for x.
In this case, the equation x^2 + 3x - 4 ≡ 0 (mod 5) factors nicely:
(x - 1)(x + 4) ≡ 0 (mod 5)
Step 5: Solve for x
To find the solutions, we need to solve each factor of the equation:
First factor: (x - 1) ≡ 0 (mod 5)
This implies x ≡ 1 (mod 5)
Second factor: (x + 4) ≡ 0 (mod 5)
This implies x ≡ -4 ≡ 1 (mod 5)
Both factors give the same solution modulo 5, so we have x ≡ 1 (mod 5).
Therefore, the solutions to the congruence relation 3x^2 + 4x - 2 ≡ 0 (mod 5) are x ≡ 1 (mod 5).