Find the zeroes of the quadratic polynomial 4x^2-6-0x and verify the relationship between the zeroes and the cofficient of the polynomial?
Answers
Answer:
The roots are x=\frac{2+\sqrt{10}}{2},\frac{2-\sqrt{10}}{2}x=
2
2+
10
,
2
2−
10
.
Step-by-step explanation:
Given : Quadratic polynomial 4x^2-6-8x4x
2
−6−8x
To find : The zero of the quadratic polynomial and verify the relationship between the zeroes and the coefficients of the polynomial ?
Solution :
Let \alpha,\betaα,β are the roots of the equation.
The polynomial in equation form, 4x^2-8x-6=04x
2
−8x−6=0
2x^2-4x-3=02x
2
−4x−3=0
Here, a=2, b=-4 and c=-3.
Using quadratic formula, x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}x=
2a
−b±
b
2
−4ac
x=\frac{-(-4)\pm\sqrt{(-4)^2-4(2)(-3)}}{2(2)}x=
2(2)
−(−4)±
(−4)
2
−4(2)(−3)
x=\frac{4\pm\sqrt{40}}{4}x=
4
4±
40
x=\frac{4\pm2\sqrt{10}}{4}x=
4
4±2
10
x=\frac{2\pm\sqrt{10}}{2}x=
2
2±
10
The roots are \alpha=\frac{2+\sqrt{10}}{2},\beta=\frac{2-\sqrt{10}}{2}α=
2
2+
10
,β=
2
2−
10
The sum of roots \alpha+\beta=-\frac{b}{a}α+β=−
a
b
\frac{2+\sqrt{10}}{2}+\frac{2-\sqrt{10}}{2}=-\frac{-4}{2}
2
2+
10
+
2
2−
10
=−
2
−4
\frac{2+\sqrt{10}+2-\sqrt{10}}{2}=2
2
2+
10
+2−
10
=2
\frac{4}{2}=2
2
4
=2
2=22=2
Verified.
The product of roots \alpha\beta=\frac{c}{a}αβ=
a
c
(\frac{2+\sqrt{10}}{2})(\frac{2-\sqrt{10}}{2})=\frac{-3}{2}(
2
2+
10
)(
2
2−
10
)=
2
−3
\frac{2^2-\sqrt{10}^2}{4}=-\frac{3}{2
\frac{4-10}{4}=-\frac{3}{2
\frac{-6}{4}=-\frac{3}{2
-\frac{3}{2}=-\frac{3}{2
Verified.