Solve this equation. 2/5 (x^2 + 1) = x
2(x^2+1) = 5x
2x^2 - 5x + 2 = 0
hint: 2*2 + 1 = 5
To solve the equation 2/5 (x^2 + 1) = x, we need to isolate the variable x on one side of the equation.
First, let's simplify the equation by distributing 2/5 to both terms inside the parentheses:
2/5 (x^2 + 1) = x
(2/5) * x^2 + (2/5) * 1 = x
(2/5) * x^2 + 2/5 = x
Now, let's move all terms to one side of the equation by subtracting x from both sides:
(2/5) * x^2 + 2/5 - x = 0
To solve this quadratic equation, we need to factor it or use the quadratic formula. In this case, let's use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation (2/5) * x^2 + 2/5 - x = 0, we have:
a = 2/5
b = -1
c = 2/5
Now we can substitute these values into the quadratic formula and solve for x:
x = (-(-1) ± √((-1)^2 - 4 * (2/5) * (2/5))) / (2 * (2/5))
x = (1 ± √(1 - 4 * (2/5) * (2/5))) / (2/5)
x = (1 ± √(1 - (16/25))) / (2/5)
x = (1 ± √(9/25)) / (2/5)
x = (1 ± 3/5) / (2/5)
We have two solutions:
x = (1 + 3/5) / (2/5) = (8/5) / (2/5) = 8/5 * 5/2 = 8/2 = 4
x = (1 - 3/5) / (2/5) = (2/5) / (2/5) = 2/5 * 5/2 = 2/2 = 1
Therefore, the solutions to the equation 2/5 (x^2 + 1) = x are x = 4 and x = 1.