Which of the following are the coordinates of the roots of x2+4x=0?
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HOMEWORK HELP > MATH
What are the roots of the equation x^2-4x+3=0. Work on two methods.
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Student Answers
WILLIAM1941 | STUDENT
We can solve the given equation using the two methods:
For the first,
x^2 - 4x + 3 = 0
express -4 as the sum of two numbers which give the product 3
=> x^2 - 3x -x + 3 = 0
=> x(x - 3) -1(x -3) =0
=> (x-1)(x-3) =0
(x -1) =0 => x= 1
and (x-3)= 0 => x = 3
So we get the roots 1 and 3
Also we can use the formula for calculating roots directly
x= [ -b + sqrt[ b^2- 4ac]/ 2a and x= [ -b - sqrt[ b^2- 4ac]/ 2a
Here a= 1, b = -4 and c=3
x= [ -b + sqrt[ b^2- 4ac]/ 2a
= [ 4 + sqrt{(-4)^2- 4*3}]/ 2
= [ 4 + sqrt 4 ]/ 2
= [ 4 + 2 ]/2
= 3
x= [ -b - sqrt[ b^2- 4ac]/ 2a
= [ 4 - sqrt{(-4)^2- 4*3}]/ 2
= [ 4 - sqrt 4 ]/ 2
= [ 4 - 2 ]/2
= 1
Again the roots are 1 and 3
So the roots we get are 1 and 3.
NEELA | STUDENT
To find the roots in 2 methods: x^2-4x+3 = 0
Let x1 and x2 be the roots.
x^2-4x+3 = k(x-x1)(x-x2) are identically equal by remainder theorem.
When k = 1, both sides have the same leading terms x^2 with 1 as coefficients. So we equate the coefficients of like powers of x and constant terms on both sides of:-
x^2-4x+3 = x^2-(x1+x2)x +x1x2.
Coefficient of x:
-4 = -( x1+x2). Or
x1+x2 = 4.........(1)
x1x2 = 3
Therefore x1-x2 = sqrt{ (x1+x2)^2 -4x1x2)} = sqrt(4^2 -4*3 )= sqrt4 = 2.
x1-x2 = 2 ..........(2)
x1+x2 = 4.........(1)
Adding the equations (1) and (2): 2x1 = 6, x 1 = 3
Subtracting eq(2) from eq(1), we get: 2x2 = 4-2 = 2. So, x2 = 1
x1 = 3 and x2 = 1.
2nd method:
x^2 -4x+3 = 0
x^2-4x = -3
Add 2^2 =4 to both sides so that LHS is a perfect square of (x-2)^2.
x^2-4x+ 4 = 4-3
(x-2)^2 = 1
Take square root:
x-2 = 1 or x-2 = -1
x = 2+1 = 3.
x = 2-1 = 1.