94. The sum of the roots of a quadratic equation is 27 and the product of its roots is 180. Find the sum of the square of its 8 0544 C 0289
Answers
Step-by-step explanation:
Let the first number be x then the second number will be 27−x.
$x(27-x)=182$$
⇒27x−x2=182
⇒x2−27x+182=0
⇒x2−13x−14x+182=0
⇒x(x−13)−14(x−13)=0
⇒(x−14)(x−13)=0
⇒x=14,13
If the first number is 14, then the second number is 13 and if the first number is 13, then the second number is 14.
Sum of the square of roots is 369
Given:
The sum of the roots of a quadratic equation is 27
Product of the roots = 180
To find:
Find the sum of the square of roots
Solution:
Let α and β be the roots of the quadratic equation
From given data sum of α, β; (α + β) = 27
Product of α, β; αβ = 180
Therefore, The quadratic equation is x² - (α + β)x + αβ = 0
The given quadratic equation is
⇒ x² - 27x + 180 = 0
Now find roots of x² - 27x + 180 = 0
As we know x = x = [-b ± √(b² – 4ac)]/2a
⇒ x = -(-27)±√( (-27)²- 4.1.180) / 2.1
= 27 ± √(729 - 720) / 2
= [27 ± √9]/2
= (27± 3)/2
⇒ x = (27 +3)/2 or x = (27 - 3)/2
x = 30/2 x = 24/2
x = 15 x = 12
Sum of the square of roots = (15)² + (12)² = 369
Sum of the square of roots is 369
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