Solve the following equation.
4m^2 = 2 − 7m
m = 1 (smaller value)
m = 2 (larger value)
To solve the equation 4m^2 = 2 - 7m, follow these steps:
1. Start by moving all the terms to one side of the equation to have a quadratic equation equal to zero:
4m^2 + 7m - 2 = 0.
2. Next, check if you can factor the quadratic equation. If you can, it makes solving much easier. However, in this case, factoring will not work.
3. To solve the equation, we can use the quadratic formula, which is:
m = (-b ± √(b^2 - 4ac)) / (2a).
In the given equation, a = 4, b = 7, and c = -2. Plug in these values into the quadratic formula:
m = (-7 ± √(7^2 - 4(4)(-2))) / (2(4)).
4. Simplify the equation inside the square root:
m = (-7 ± √(49 + 32)) / 8.
m = (-7 ± √81) / 8.
m = (-7 ± 9) / 8.
5. Calculate both potential solutions:
Solution 1: m = (-7 + 9) / 8 = 2 / 8 = 1/4 = 0.25.
Solution 2: m = (-7 - 9) / 8 = -16 / 8 = -2.
Therefore, the solutions to the equation 4m^2 = 2 - 7m are:
m = 0.25 (or 1/4) as the smaller value,
m = -2 as the larger value.