Start by identifying the values of a, b, and c. Be sure that all of the terms are on the same side and in the correct order for the standard form of a quadratic function.
3/4 and - 4/5
Practice makes perfect
To solve the given equation by factoring, we will start by identifying the values of a, b, and c.
40x^2+2x= 24 ⇔ 40x^2+ 2x+( - 24)=0
We have a quadratic equation with a= 40, b= 2, and c= - 24. To factor the left-hand side, we need to find a factor pair of 40 * - 24=- 960 whose sum is 2.
Since - 960 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 - 960
^(1* (- 960)) - 960
1+(- 960) - 959
- 1 and 960
^(- 1* 960) - 960
- 1+960 959
2 and - 480
^(2* (- 480)) - 960
2+(- 480) - 478
- 2 and 480
^(- 2* 480) - 960
- 2+480 478
3 and - 320
^(3* (- 320)) - 960
3+(- 320) - 317
- 3 and 320
^(- 3* 320) - 960
- 3+320 317
4 and - 240
^(4* (- 240)) - 960
4+(- 240) - 236
- 4 and 240
^(- 4* 240) - 960
- 4+240 236
5 and - 192
^(5* (- 192)) - 960
5+(- 192) - 187
- 5 and 192
^(- 5* 192) - 960
- 5+192 187
6 and - 160
^(6* (- 160)) - 960
6+(- 160) - 154
- 6 and 160
^(- 6* 160) - 960
- 6+160 154
8 and - 120
^(8* (- 120)) - 960
8+(- 120) - 112
- 8 and 120
^(- 8* 120) - 960
- 8+120 112
10 and - 96
^(10* (- 96)) - 960
10+(- 96) - 86
- 10 and 96
^(- 10* 96) - 960
- 10+96 86
12 and - 80
^(12* (- 80)) - 960
12+(- 80) - 68
- 12 and 80
^(- 12* 80) - 960
- 12+80 68
15 and - 64
^(15* (- 64)) - 960
15+(- 64) - 49
- 15 and 64
^(- 15* 64) - 960
- 15+64 49
16 and - 60
^(16* (- 60)) - 960
16+(- 60) - 44
- 16 and 60
^(- 16* 60) - 960
- 16+60 44
20 and - 48
^(20* (- 48)) - 960
20+(- 48) - 28
- 20 and 48
^(- 20* 48) - 960
- 20+48 28
24 and - 40
^(24* (- 40)) - 960
24+(- 40) - 16
- 24 and 40
^(- 24* 40) - 960
- 24+40 16
30 and - 32
^(30* (- 32)) - 960
30+(- 32) - 2
- 30 and 32
^(- 30* 32) - 960
- 30+32 2
The integers whose product is - 960 and whose sum is 2 are - 30 and 32. 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= 34 and x=- 45. To check our answer we will graph the related function y=40x^2+2x-24 using a calculator.
We can see that the x-intercepts are 0.75, or 34, and - 0.8, or - 45. Therefore, our solutions are correct.
