find the zeros of a quadratic polynomial 4 x square + 5 root 2 x minus 3 and verify the relationship between zero and its coefficient
Answers
Answer:
Step-by-step explanation:
4x
2
+5
2
x−3
=4x
2
+6
2
x−
2
x−3
=2
2
x(
2
x+3)−(
2
x+3)
=(
2
x+3)(2
2
x−1)
So, the zeros of the polynomial are
(
2
x+3)=0 and (2
2
x−1)=0
x=
2
−3
and x=
2
2
1
x=
2
−3
2
and x=
4
2
Now, the polynomial is 4x
2
+5
2
x−3
Comparing it with ax
2
+bx+c=0, we get a=4, b=5
2
and c=−3
Sum of the zeros =
2
−3
2
+
4
2
=
4
−6
2
+
2
=
4
−5
2
=
a
−b
Product of the zeros =(
2
−3
2
)(
4
2
)
=(
2
−3
)(
4
2
)
=
4
−3
=
a
c
Solution:
Given Equation –
→ 4x² + 5√2x - 3 = 0
By splitting the middle term, we get,
→ 4x² + 6√2x - √2x - 3 = 0
→ 2√2x(√2x + 3) - 1(√2x + 3) = 0
→ (2√2x - 1)(√2x + 3) = 0
By zero product rule,
→ (2√2x - 1) = 0 or (√2x + 3) = 0
→ x = 1/(2√2) and -3/√2
Verification:
Given Equation –
→ 4x² + 5√2x - 3 = 0
Comparing with ax² + bx + c = 0, we get,
→ a = 4
→ b = 5√2
→ c = -3
We know that,
→ Sum of Roots = -b/a
→ Product of Roots = c/a
Here,
→ -b/a = -5√2/4
→ c/a = -3/4
Again,
1/(2√2) + (-3/√2)
= 1/2√2 - 6/2√2
= (1 - 6)/2√2
= -5/2√2
Therefore,
→ Sum of roots = -b/a (Verified)
Now,
1/2√2 × -3/√2
= -3/4
Therefore,
→ Product of roots = c/a (Verified)