solve step by step
0=-2(8t^2-32t+21)
-16t^2 + 64t - 42 = 0
Divide end values into two factors (-7,6 or 7,-6, or 21,-2 or -21,2 or 14, -3 or so on) (16,-1 or -16,1 or 4,-4 or -8, 2 and so on) to see how their sum will get the center value.
From there it is trial and error. Start trying.
To solve the equation step by step, we can start by simplifying the equation and then isolating the variable.
Given: 0 = -2(8t^2 - 32t + 21)
Step 1: Distribute the -2 to each term inside the parentheses.
0 = -16t^2 + 64t - 42
Step 2: Rearrange the equation in standard quadratic form, which is ax^2 + bx + c = 0.
-16t^2 + 64t - 42 = 0
Step 3: Divide the entire equation by -2 to simplify the equation further (optional step).
8t^2 - 32t + 21 = 0
Now, we can move on to solving the quadratic equation. There are several methods, but I will explain how to solve it using the quadratic formula.
Step 4: Identify the values of a, b, and c in the quadratic equation.
In our equation, a = 8, b = -32, and c = 21.
Step 5: Substitute the values of a, b, and c into the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Step 6: Plug in the values into the formula and simplify.
t = (-(-32) ± √((-32)^2 - 4 * 8 * 21)) / (2 * 8)
t = (32 ± √(1024 - 672)) / 16
t = (32 ± √(352)) / 16
Step 7: Simplify the square root if possible.
t = (32 ± √(16 * 22)) / 16
t = (32 ± 4√22) / 16
Step 8: Simplify the expression further.
t = 2(8 ± √22) / 16
t = (8 ± √22) / 8
Finally, we can simplify the expression further by dividing numerator and denominator by 8 to get:
t = 1 ± (√22 / 4)
So, the solutions to the equation 0 = -2(8t^2 - 32t + 21) are t = 1 + (√22 / 4) and t = 1 - (√22 / 4).