Answer:
t = (5 + √21)/4 and t = (5 - √21)/4.
Step-by-step explanation:
To solve the equation 4t^2 - 5t + 1 = 0, we will first try to factor the expression:
1. To factor the quadratic equation 4t^2 - 5t + 1 = 0, we need to find two numbers that multiply to 4 1 = 4 (the coefficient of t^2 the constant term) and add up to -5 (the coefficient of t). These numbers are -4 and -1. So, we rewrite the equation as:
(4t - 1)(t - 1) = 0.
2. Setting each factor to zero, we solve for t:
4t - 1 = 0 and t - 1 = 0.
3. Solving for t in each equation, we get:
4t - 1 = 0 -> 4t = 1 -> t = 1/4, and
t - 1 = 0 -> t = 1.
Therefore, the solutions for the equation 4t^2 - 5t + 1 = 0 when factored are t = 1/4 and t = 1.
If factoring is not possible, we can solve the quadratic equation by completing the square:
4t^2 - 5t + 1 = 0.
1. Move the constant to the other side:
4t^2 - 5t = -1.
2. To complete the square, halve the coefficient of t, square it, and add it to both sides:
4t^2 - 5t + (-5/2)^2 = -1 + (-5/2)^2,
4t^2 - 5t + 25/4 = -1 + 25/4,
4t^2 - 5t + 25/4 = 21/4.
3. Rewrite the left side as a perfect square:
(2t - 5/2)^2 = 21/4.
4. Take the square root of both sides:
2t - 5/2 = ±√21/2.
5. Solve for t:
2t = 5/2 ± √21/2,
t = (5 ± √21)/4.
Therefore, the solutions for the equation 4t^2 - 5t + 1 = 0 by completing the square are t = (5 + √21)/4 and t = (5 - √21)/4.
