2x−5√x+2=0
quadratic equations
let √x = y
then:
2y^2 - 5y + 2 = 0
(2y -1)(y -2) = 0
y = 1/2 or y = 2
√x = 1/2 ----> x = 1/4
or
√x = 2 ------> x = 4
or
2x+2 = 5√x
square both sides:
4x^2 + 8x + 4 = 25x
4x^2 - 17x + 4 = 0
(x - 4)(4x - 1) =0
x = 4 or x = 1/4
btw, in both solutions, my answers have to verified since we used squaring the the process.
Luckily both answers work.
Ty! :)
or, just factor it:
(√x-2)(2√x-1) = 0
√x = 2 or 1/2
To solve the quadratic equation 2x - 5√x + 2 = 0, we can follow these steps:
Step 1: Make a substitution
Let's say u = √x. This substitution will help us simplify the equation.
So, the equation becomes: 2u^2 - 5u + 2 = 0.
Step 2: Solve the quadratic equation
Now, we have a quadratic equation in terms of u. We can solve it by factoring, completing the square, or using the quadratic formula.
Since factoring may not be straightforward in this case, let's use the quadratic formula:
The quadratic formula is given by: u = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation.
Applying the formula to our equation, we have:
u = (-(-5) ± √((-5)^2 - 4(2)(2))) / (2(2))
Simplifying further, we get:
u = (5 ± √(25 - 16)) / 4
u = (5 ± √9) / 4
u = (5 ± 3) / 4
So, we have two possible values for u:
u1 = (5 + 3) / 4 = 8 / 4 = 2
u2 = (5 - 3) / 4 = 2 / 4 = 1/2
Step 3: Find the corresponding values of x
Now, we need to find the corresponding values of x based on our substitution.
For u1 = 2:
u = √x
√x = 2
x = (2)^2
x = 4
For u2 = 1/2:
u = √x
√x = 1/2
x = (1/2)^2
x = 1/4
So, the solutions to the quadratic equation 2x - 5√x + 2 = 0 are:
x1 = 4
x2 = 1/4