Final answer:
To solve the quadratic equation 6x² -41x+30=0 by factoring, we first find two numbers that multiply to 180 and sum to -41, which are -36 and -5. We then use these numbers to factor by grouping and obtain (6x - 5)(x - 6) = 0. The solutions are x = 5/6 and x = 6.
Step-by-step explanation:
To solve 6x² -41x+30=0 by factoring, we need to find two numbers that multiply to give the product of the coefficient of x² (which is 6) and the constant term (which is 30) and add up to give the coefficient of x (which is -41).
The product of 6 and 30 is 180. We are looking for two numbers that multiply to 180 and add up to -41. These numbers are -36 and -5 because (-36) * (-5) = 180 and (-36) + (-5) = -41. Now we can rewrite the middle term of the equation using these two numbers:
6x² -36x -5x + 30 = 0
Next, we can factor by grouping:
(6x² -36x) + (-5x + 30) = 0
6x(x - 6) - 5(x - 6) = 0
Now we can factor out the common binomial factor (x - 6):
(6x - 5)(x - 6) = 0
Finally, by the zero product property, we set each factor equal to zero and solve for x:
- 6x - 5 = 0 → x = 5/6
- x - 6 = 0 → x = 6
Therefore, the solutions are x = 5/6 and x = 6.