Question

Write a quadratic equation if its roots are: 2-√3, and 1/(2-√3) assuming that the leading coefficient a=1, the equation is...

Answer 1

To write a quadratic equation with given roots, use the fact that the sum and product of the roots are related to the coefficients of the equation.

Step-by-step explanation:

To write a quadratic equation with roots 2-√3 and 1/(2-√3), we can use the fact that the sum and product of the roots of a quadratic equation are related to the coefficients of the equation.

Let's assume the equation is ax² + bx + c = 0, where a = 1. We know that the sum of the roots is equal to -b/a and the product of the roots is equal to c/a. So we can set up the following equations:

• 2 - √3 + 1/(2 - √3) = -b/1
• (2 - √3) * 1/(2 - √3) = c/1

Simplifying these equations, we can find the values of b and c, and then plug them into the quadratic equation.

Answer 2

Writing a Quadratic Equation given its Roots

Answer:

`x^2 -4x +1 = 0`

Step-by-step explanation:

According to the Zero-Product Property, if you have the Quadratic Equation,
`(x +a)(x +b) =0`, it's roots will be
`x = -h` and
`x = -k`.

If we want our roots to be
`2 -\sqrt 3` and
`(1)/(2 -\sqrt 3)\\`, then
`-h = 2 -\sqrt 3` and
`-k = (1)/(2 -\sqrt 3)\\`.

Solving for
`h`:

`-h = 2 -\sqrt 3 \\ -h \cdot -1 = (2 -\sqrt 3) \cdot -1 \\ h = \sqrt 3 -2`

Solving for
`k`:

`-k = (1)/(2 -\sqrt 3) \\ -k \cdot -1 = (1)/(2 -\sqrt 3) \cdot -1 \\ k = (1)/(\sqrt 3 -2)`

Writing the equation:

`(x +h)(x +k) = 0 \\ (x +(\sqrt 3 -2))(x +(1)/(\sqrt 3 -2)) = 0`

Unsurprisingly, the answer shouldn't be that because it shouldn't be that easy. By that, we have to apply the distributive property.

`(x +(\sqrt 3 -2))(x +(1)/(\sqrt 3 -2)) = 0 \\ x(x +(\sqrt 3 -2)) +(1)/(\sqrt 3 -2)(x +(\sqrt 3 -2)) = 0 \\ x^2 +(\sqrt 3 -2)x +(1)/(\sqrt 3 -2)x +(1)/(\sqrt 3 -2)(\sqrt 3 -2) = 0 \\ x^2 +((\sqrt 3 -2) +(1)/(\sqrt 3 -2))x +1 = 0 \\ x^2 + (((\sqrt 3 -2)^2)/(\sqrt 3 -2) +(1)/(\sqrt 3 -2))x +1 = 0 \\ x^2 +(3 -4\sqrt 3 +4 +1)/(\sqrt 3 -2)x +1 = 0 \\ x^2 +(8 -4\sqrt 3)/(\sqrt 3 -2)x +1 = 0 \\ x^2 +(-4(\sqrt 3 -2))/(\sqrt 3 -2)x +1 = 0 \\ x^2 -4x +1 = 0`