Show that the equation 2x^2 - 5x + 2 = 0 has roots that are reciprocals of each other. Under what conditions will a quadratic equation in the form ax ^2 +bx + c = 0 have roots that are reciprocals of each other?
factor out 2 above:
2(x^2-2.5x +1)=0
Now, how does to values of x multiplied become 1 ?
To show that the equation 2x^2 - 5x + 2 = 0 has roots that are reciprocals of each other, we need to find the roots of the equation and demonstrate that they are reciprocals.
First, let's find the roots of the equation. We can use the quadratic formula to find the solutions of any quadratic equation in the form ax^2 + bx + c = 0.
The quadratic formula states that for an equation ax^2 + bx + c = 0, the solutions (or roots) are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation 2x^2 - 5x + 2 = 0, we have a = 2, b = -5, and c = 2. Substituting these values into the quadratic formula, we get:
x = (-(-5) ± √((-5)^2 - 4(2)(2))) / (2(2))
= (5 ± √(25 - 16)) / 4
= (5 ± √9) / 4
Simplifying further, we have:
x = (5 ± 3) / 4
This gives us two possible values for x:
x₁ = (5 + 3) / 4 = 8 / 4 = 2
x₂ = (5 - 3) / 4 = 2 / 4 = 1/2
Therefore, the roots of the equation 2x^2 - 5x + 2 = 0 are x = 2 and x = 1/2.
To determine if these roots are reciprocals of each other, we can calculate the reciprocal of each root:
Reciprocal of 2: 1/2
Reciprocal of 1/2: 2
As we can see, the reciprocals of the roots 2 and 1/2 are indeed each other. Therefore, the equation 2x^2 - 5x + 2 = 0 has roots that are reciprocals of each other.
Now, let's discuss the conditions for a quadratic equation in the form ax^2 + bx + c = 0 to have roots that are reciprocals of each other.
For a quadratic equation to have roots that are reciprocals of each other, the sum of the roots should be zero.
Using Vieta's formulas, the sum of the roots of a quadratic equation ax^2 + bx + c = 0 is given by:
Sum of roots = -b/a
For the roots to be reciprocals, the sum of the roots should be zero. Therefore, we need:
-b/a = 0
Since the denominator cannot be zero, the condition for a quadratic equation to have roots that are reciprocals of each other is b = 0.
In summary, for a quadratic equation ax^2 + bx + c = 0 to have roots that are reciprocals of each other, the coefficient of the linear term (b) must be zero.