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.
1/4 and 2
Practice makes perfect
To solve the given equation by factoring, we will start by identifying the values of a, b, and c.
- 16 t^2+36t-8=0 ⇔ - 16t^2+ 36t+( - 8)=0
We have a quadratic equation with a= - 16, b= 36, and c= - 8. To factor the left-hand side we need to find a factor pair of - 16 * - 8=128 whose sum is 36.
Since 128 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 128
^(1* 128) 128
1+128 129
- 1 and - 128
^(- 1* (- 128)) 128
- 1+(- 128) - 129
2 and 64
^(2* 64) 128
2+64 66
- 2 and - 64
^(- 2* (- 64)) 128
- 2+(- 64) - 66
4 and 32
^(4* 32) 128
4+32 36
- 4 and - 32
^(- 4* (- 32)) 128
- 4+(- 32) - 36
The integers whose product is 128 and whose sum is 36 are 4 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 t=2 and t= 14. To check our answer, we will graph the related function y=- 16t^2+36t-8 using a calculator. Note that in the calculator we will use the variable x instead of t.
We can see that the x-intercepts are 0.25, or 14, and 2. Therefore, the solutions are correct. ✓
