Question

find the zeros of following quadratic polynomial and verify the relationship between the zeros and the coefficient of the polynomial f(x)=5x-4√3+2√3x²​

Answer 1

Explanation:

Rewrite the given polynomial in the form ax² + bx + c:

To find the zeros, set the function to zero and solve for x using the quadratic formula.

Quadratic formula:

Therefore,

a = 2√3

b = 5

c = - 4√3

Substituting the values into the quadratic formula:

The sum of the roots of a polynomial is -b/a:

The sum of the found roots is:

Hence proving the sum of the roots is -b/a

The product of the roots of a polynomial is: c/a

The product of the found roots is:

Hence proving the product of the roots is c/a

Therefore, the relationship between the roots and the coefficients is verified.

Answer 2

`\textsf{Zeros}: \quad x=\frac {√(3)}{2}, \:\:x=-\frac {4√(3)}{3}`

Explanation:

Rewrite the given polynomial in the form ax² + bx + c:

`f(x)=2 √(3)x^2+5x-4 √(3)`

To find the zeros, set the function to zero and solve for x using the quadratic formula.

`\implies 2 √(3)x^2+5x-4 √(3)=0`

Quadratic formula:

`x=(-b \pm √(b^2-4ac) )/(2a)\quad\textsf{when }\:ax^2+bx+c=0`

Therefore,

• a = 2√3
• b = 5
• c = - 4√3

Substituting the values into the quadratic formula:

`\implies x=\frac{-5 \pm \sqrt{5^2-4(2√(3))(-4√(3))} }{2(2√(3))}`

`\implies x=\frac {-5 \pm \sqrt {121}}{4√(3)}`

`\implies x=\frac {-5 \pm 11}{4√(3)}`

`\implies x=\frac {6}{4√(3)}, \:\:x=\frac {-16}{4√(3)}`

`\implies x=\frac {3}{2√(3)}, \:\:x=-\frac {4}{√(3)}`

`\implies x=\frac {√(3)}{2}, \:\:x=-\frac {4√(3)}{3}`

The sum of the roots of a polynomial is -b/a:

`\implies -(b)/(a)=-(5)/(2 √(3))=-(5√(3))/(6)`

The sum of the found roots is:

`\implies \left(\frac {√(3)}{2}\right)+\left(-\frac {4√(3)}{3}\right)=-(5√(3))/(6)`

Hence proving the sum of the roots is -b/a

The product of the roots of a polynomial is: c/a

`\implies (c)/(a)=(-4√(3))/(2√(3))=-2`

The product of the found roots is:

`\implies \left(\frac {√(3)}{2}\right)\left(-\frac {4√(3)}{3}\right)=-(12)/(6)=-2`

Hence proving the product of the roots is c/a

Therefore, the relationship between the roots and the coefficients is verified.