Find the roots of the following quadratic equations, if they exist by the method of completing the square:
x2 – 9x + 18 = 0
Answers
Question:
Find the roots of the following quadratic equations, if they exist by the method of completing the square: x² - 9x + 18 = 0
Answer:
x = 4 , 6
Note:
• The possible values of unknown (variable) for which the equation is satisfied are called its solutions or roots .
• If x = a is a solution of any equation in x , then it must satisfy the given equation otherwise it's not a solution (root) of the equation.
• The discriminant of the the quadratic equation
ax² + bx + c = 0 , is given as ; D = b² - 4ac
• If D > 0 then its roots are real and distinct.
• If D < 0 then its roots are imaginary.
• If D = 0 then its roots are real and equal.
Solution:
Here,
The given quadratic equation is :
x² - 9x + 18 = 0
Clearly, here we have ;
a = 1
b = -9
c = 18
Now,
The discriminant will be ;
=> D = b² - 4ac
=> D = (-9)² - 4•1•18
=> D = 81 - 72
=> D = 9. (D > 0)
Since,
The discriminant of the given quadratic equation is greater than zero, thus there must exist two distinct roots.
Now,
=> x² - 9x + 18 = 0
=> x² - 9x + (9/2)² - (9/2)² + 18 = 0
=> x² - 2•x•(9/2) + (9/2)² = (9/2)² - 18
=> (x - 9/2)² = 81/4 - 18
=> (x - 9/2)² = (81 - 72)/4
=> (x - 9/2)² = 9/4
=> x - 9/2 = √(9/4)
=> x - 9/2 = ± 3/2
=> x = 9/2 ± 3/2
Case1
=> x = 9/2 + 3/2
=> x = (9+3)/2
=> x = 12/2
=> x = 6
Case2
=> x = 9/2 - 3/2
=> x = (9-3)/2
=> x = 6/2
=> x = 4
Hence,
The required roots of the given quadratic equation are : x = 4 , 6 .
Step-by-step explanation:
x²-6x-3x+18=0
x(x-6)-3(x-6)=0
(x-6)(x-3)=0
X=6 or 3
no need completing square.