prove that if m is not equal to n, the equation (m^2+n^2)x^2 - 2(m+n)x + 2=0 has unreal roots.
In quadratic equation:
a x ^ 2 + b x + c
expresion :
delta = b ^ 2 - 4 a c
is called the Discriminant
When delta > 0
There is two real root
When delta = 0
There is 2 equal real roots. ( The root is repeated. )
When delta < 0
There are two complex ( unreal ) roots.
In this case :
a = m ^ 2 + n ^ 2
b = - 2 ( m + n )
c = 2
delta = b ^ 2 - 4 ac
delta = [ - 2 ( m + n ) ] ^ 2 - 4 * ( m ^ 2 + n ^ 2 ) * 2
delta = 4 * ( m + n ) ^ 2 - 8 * ( m ^ 2 + n ^ 2 )
delta = 4 * ( m ^ 2 + 2 m n + n ^ 2 ) - 8 m ^ 2 - 8 n ^ 2
delta = 4 m ^ 2 + 8 m n + 4 n ^ 2 - 8 m ^ 2 - 8 n ^ 2
delta = - 4 m ^ 2 + 8 m n - 4 n ^ 2
delta = - 4 ( m ^ 2 - 2 m n + n )
delta = - 4 ( m - n ) ^ 2
Becouse : ( m - n ) ^ 2 = m ^ 2 - 2 * m n + n ^ 2
So delta = - 4 ( m - n ) ^ 2
When m > n
( m - n ) are positive
Square of positive number are positive number.
- 4 * positive number are negative number
When m < n
( m - n ) are negative
Square of negative number are positive number.
- 4 * positive number are negative number
This mean when m is not equal to n the Disciminant is always negative , and your equation always have two unreal roots.
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Remark :
delta is greek letter delta ( Like triangle).
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To prove that the equation (m^2+n^2)x^2 - 2(m+n)x + 2=0 has unreal roots when m is not equal to n, we can use the discriminant of a quadratic equation.
The discriminant (D) is a value used to determine the nature of the roots of a quadratic equation. It is calculated using the formula:
D = b^2 - 4ac
In this case, the quadratic equation is in the form ax^2 + bx + c = 0, with a = m^2 + n^2, b = -2(m + n), and c = 2.
Substituting these values into the discriminant formula, we get:
D = (-2(m + n))^2 - 4(m^2 + n^2)(2)
D = 4(m + n)^2 - 8(m^2 + n^2)
D = 4m^2 + 8mn + 4n^2 - 8m^2 - 8n^2
D = -4m^2 - 8n^2 + 8mn
Since we are trying to prove that the roots are unreal, we want to show that the discriminant (D) is negative. So, in order to prove this, we need to show that -4m^2 - 8n^2 + 8mn < 0.
Now, let's consider the given condition that m is not equal to n. Since m and n are distinct numbers, we can assume that mn ≠ 0. With this assumption, we can divide the expression -4m^2 - 8n^2 + 8mn by -4mn to simplify the inequality:
(-4m^2 - 8n^2 + 8mn)/(-4mn) < 0
Canceling out the negative signs, we get:
(m^2 + 2n^2 - 2mn)/(mn) > 0
Now, let's analyze the numerator of this fraction:
m^2 + 2n^2 - 2mn
If we factor out a 2 from the last two terms:
m^2 + 2(n^2 - mn)
Since m ≠ n, we know that n^2 - mn > 0 (subtracting mn from n^2 results in a positive number). Therefore, the numerator is positive.
Since the numerator is positive, and the denominator is also positive (since we assumed mn ≠ 0), the entire fraction is positive. Therefore, we have shown that:
(m^2 + 2n^2 - 2mn)/(mn) > 0
This means that -4m^2 - 8n^2 + 8mn < 0, which implies that the discriminant (D) is negative. Since the discriminant is negative, the quadratic equation (m^2+n^2)x^2 - 2(m+n)x + 2=0 has unreal roots.
Therefore, we have proven that if m is not equal to n, the equation (m^2+n^2)x^2 - 2(m+n)x + 2=0 has unreal roots.