Start by identifying the values of a, b, and c. Make sure that all of the terms are on the same side and in the correct order for the standard form of a quadratic function.
- 5/3 and - 4
Practice makes perfect
To solve the given equation by factoring, we will start by identifying the values of a, b, and c.
3x^2+ 17x+ 20=0
We have a quadratic equation with a= 3, b= 17, and c= 20. To factor the left-hand side, we need to find a factor pair of 3 * 20=60 whose sum is 17.
Since 60 is a positive number, we will only consider factors with the same sign — both positive or both negative — so that their product is positive.
Factor Pair
Product of Factors
Sum of Factors
1 and 60
^(1* 60) 60
1+60 61
- 1 and - 60
^(- 1* (- 60)) 60
- 1+ (- 60) - 61
2 and 30
^(2* 30) 60
2+30 32
- 2 and - 30
^(- 2* (- 30)) 60
- 2+ (- 30) - 32
3 and 20
^(3* 20) 60
3+ 20 23
- 3 and - 20
^(- 3* (- 20)) 60
- 3+ (- 20) - 23
4 and 15
^(4* 15) 60
4+ 15 19
- 4 and - 15
^(- 4* (- 15)) 60
- 4+(- 15) - 19
5 and 12
^(5* 12) 60
5+12 17
- 5 and - 12
^(- 5* (- 12)) 60
- 5+ (- 12) - 17
6 and 10
^(6* 10) 60
6+ 10 16
- 6 and - 10
^(- 6* (- 10)) 60
- 6+ (- 10) - 16
The integers whose product is 60 and whose sum is 17 are 12 and 5. With this information, we can rewrite the linear factor on the left-hand side of the equation, and factor by grouping.
We found that the solutions to the given equation are x=- 53 and x=- 4. To check our answer, we will graph the related function, y=3x^2+17x+20, using a calculator.
We can see that the x-intercepts are - 53 and - 4. Therefore, our solutions are correct.
