4x^2 +4x + 1.find sum and product of roots.?

Question

also give the formula for solving.

Answer 1

first solution)
4x^2 + 4x +1 = (2x+1)^2
x1 = x2 = -1/2
sum is -1 and product is 1/4

seccond solution)
in any parabolic equation like : Ax^2 + Bx + C = 0
sum of roots is -B/A and product of roots is C/A.
in this case :
-B/A = -4/4 = -1 and C/A = 1/4

Answer 2

This quadratic is a perfect trinomial square, and you should recognize that right away. All the stuff about factoring, quadratic formula, or complete the square is mumbo jumbo.

Look, 4x^2 is a square, and so is +1. When you see that, you try (2x + 1)^2. To check the middle term, just double the product of your two roots: 2(2x)(1) = +4x (middle term is okay).

So all this means

4x^2 + 4x + 1 = (2x + 1)^2

Set that equal to zero to get double roots

2x + 1 = 0 ==> x = -1/2 (double root)

The sum of the roots is -1/2 - 1/2 = -1.

The product of the roots is (-1/2)^2 = 1/4.

Answer 3

4x^2 + 4x + 1
First, make expression equal to 0
i.e. 4x^2 + 4x + 1 = 0

Then find two numbers when added will give (4) as in 4x and when multiplied will give 4 as in the 4 from 4x^2 multiplied by 1 (as in the +1).
The two numbers are 2 and 2

keep in mind that if the first term has a number (4) before x^2 then it should be multiplied by the last number (1) to find two numbers when added will give 4 (before x) and when multiplied will give 4 (4 from 4x^2 multiplied by 1).


rewrite the original equation using the two numbers (2 and 2)

4x^2 + 2x + 2x + 1=0

then divide the first two terms by the highest common factor (hcf) which in this case is 2x.

2x (2x + 1) + 2x + 1=0

the highest common factor for the two other terms is 1 therefore:

2x (2x + 1) + 1 (2x + 1)=0

Notice the same occurrence in the brackets.

then divide on either side by 2x + 1

2x + 1 (2x +1)=0

So since these terms are multiplied, one of them have to be zero, because anything multiplied by zero is zero.

but since the terms are the same

2x + 1 = 0
2x = 0 - 1
2x = - 1
x = -1/2

Answer 4

4x^2 + 4x + 1 = 0
move the third term to the right of the equal sign, leaving a placeholder for a new third term
4x^2 + 4x + __ = 1 + __
divide the middle term by 2; then square to determine the value of the third term
4x^2 + 4x + 4 = 1 + 4
4x^2 + 4x + 4 = 5
rewrite the left side in simplified form
(2x +2)^2 = 5
take the square root of each side
2x + 2 = plus or minus the square root of 5
2x = -2 plus or minus the square root of 5
x = (-2 plus or minus the square root of 5) / 2

x = 2.236067977 or x = -2.236067977

sum of roots = 0
product of roots = -4.999999998

Answer 5

You can use the quadratic formula. With this you will get two roots. That way you can add and multiply them together when you're done.

Quadratic formula is: (-b+/- sqrt(b^2-4ac))/(2a)

In your equation 4 is a, 4 is b, and 1 is c. (the number infront of the squared x is always a, infront of the x^1 is always b, and the number by itself is always c)

(-4+/- sqrt(4^2-4*4*1))/2*4
(-4+/- sqrt(16-16))/8
(-4+/- 0)/8
-4/8= -1/2
x=-1/2, -1/2

sum of the roots is: -1/2+-1/2=-1
product of the roots is: (-1/2)(-1/2)=1/4

Answer 6

First, x^2 should have a coefficient of 1, so:
4x^2 +4x +1 =0
x^2 +x +(1/4) =0

the sum is the coefficient of -x = -1
the product is the constant = 1/4

so a*b = 1/4
a+b = -1
x^2 +x +(1/4) =0
(x +(1/2) ) (x + (1/2) )=0
a and b are both -1/2

Answer 7

By factoring: (2x + 1)(2x + 1)
2x + 1 = 0
2x = -1
x = -1/2 <-- This is the root so...
Sum of the roots = x + x = -1/2 + -1/2 = -1
Product of the roots = x * x = -1/2 x -1/2 = 1/4

Hope this helps...

Answer 8

4x^2 + 4x + 1 = 0
1/4 (4x + 2) (4x + 2) = 0
(2x + 1)^2 = 0
(2x + 1)=0
2x = -1
x = -1/2
so, the roots are 1/2 and 1/2

the sum is... 1/2 + 1/2 = 1
the product is ... 1/2 * 1/2 = 1/4

Answer 9

Sum=4.5,product= -8.Use the quadratic formula to get the roots of x.In this case x has only 1 solution.

Answer 10

first i will solve the quadratic equation:
4x^2+4x+1 = 0
First we need to reform:
ax^2 + bx + c = 0
For this, we arrange the terms from the highest exponent to the lowest:
4x^2+4x+1 = 0
We then have:
a=4 b=4 c=1
now apply the quadratic formula:
x = ( -b ± √(b²-4ac) ) / 2a
there is one solution:
x=-0.5