Start by identifying the values of a, b, and c.
Be sure that all of the terms of are on the same side and in the correct order for the standard form of a quadratic function.
- 5/4 and 6
Practice makes perfect
To solve the given equation by factoring, we will start by identifying the values of a, b, and c.
- 4x^2+19x=- 30 ⇔ - 4x^2+ 19x+ 30=0
We have a quadratic equation with a= - 4, b= 19, and c= 30. To factor the left-hand side, we need to find a factor pair of -4 * 30=- 120 whose sum is 19.
Since - 120 is a negative number, we will only consider factors with opposite signs — one positive and one negative — so that their product is negative.
Factor Pair
Product of Factors
Sum of Factors
1 and - 120
^(1* (- 120)) - 120
1+(- 120) - 119
- 1 and 120
^(- 1* 120) - 120
- 1+120 119
2 and - 60
^(2* (- 60)) - 120
2+(- 60) - 58
- 2 and 60
^(- 2* 60) - 120
- 2+60 58
3 and - 40
^(3* (- 40)) - 120
3+(- 40) - 37
- 3 and 40
^(- 3* 40) - 120
- 3+40 37
4 and - 30
^(4* (- 30)) - 120
4+(- 30) - 26
- 4 and 30
^(- 4* 30) - 120
- 4+30 26
5 and - 24
^(5* (- 24)) - 120
5+(- 24) - 19
- 5 and 24
^(- 5* 24) - 120
- 5+24 19
6 and - 20
^(6* (- 20)) - 120
6+(- 20) - 14
- 6 and 20
^(- 6* 20) - 120
- 6+20 14
8 and - 15
^(8* (- 15)) - 120
8+(- 15) - 7
- 8 and 15
^(- 8* 15) - 120
- 8+15 7
10 and - 12
^(10* (- 12)) - 120
10+(- 12) - 2
- 10 and 12
^(- 10* 12) - 120
- 10+12 2
The integers whose product is - 120 and whose sum is 19 are - 5 and 24. 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=- 54 and x=6. To check our answer, we will graph the related function y=- 4x^2+19x+30 using a calculator.
We can see that the x-intercepts are - 1.25, or - 54, and 6. Therefore, are solutions are correct. ✓
