Move all terms not containing `x` to the right-hand side of the equation.

`4x^2-4x = -1`

To set the left-hand side of the equation equal to `0` , move all the expressions to the left-hand side.

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

Remove the fraction by multiplying the first term of the factor by the denominator of the second term.

`(2x-1)(2x-1)=0`

Set each of the factors of the left-hand side of the equation equal to `0` .

`2x-1=0`

Since `-1` does not contain the variable to solve for, move it to the right-hand side of the equation by adding `1` to both sides.

`2x=1`

Divide each term in the equation by `2` .

`x=1/2`

The equation 4x^2 = 4x - 1 has to be solved for x.

Write the equation in the standard form of a quadratic equation ax^2 + bx + c = 0

4x^2 = 4x - 1

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

The polynomial 4x^2 - 4x + 1 has to be factored and each of them equated to 0 to determine the root.

4x^2 - 4x + 1 = 0

Notice that 4x^2 = (2x)^2 and 4x = 2*2x*1. Substituting the same gives: (2x)^2 - 2*2x*1 + 1^2 = 0

The left hand side is in the form a^2 - 2*a*b + b^2 which is equal to (a - b)^2

4x^2 - 4x + 1 = 0

=> (2x - 1)^2 = 0

=> x = 1/2

**The solution of the equation 4x^2 = 4x - 1 is x = 1/2**

**QUESTION:-**

**Factor the equation to find the roots: **

**4x^2 = 4x - 1**

**SOLUTION:-**

This problem can be solved by two methods which are as follows:-

- FACTORIZATION METHOD
- QUADRATIC EQUATION FORMULA METHOD

Both the methods will give the same answer or solution set.

FACTORIZATION METHOD:-

The equation to be solved is;

`4x^2=4x-1`

Bring it in proper form which is;

ax^2+bx+c=0

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

` `

Now factorization method is to divide the middle value in such a way that the value that comes after taking common is same;

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

Take common;

`2x(2x-1)-1(2x-1)=0`

`(2x-1)(2x-1)=0`

`2x-1=0`

`2x=1`

`x=1/2`

QUADRATIC EQUATION FORMULA METHOD

The quadratic equation formula is;

`x={-b+-sqrt(b^2-4ac)}/(2a)`

According to the problem,

a = 4

b = -4

c = 1

Now insert the values of a, b and c in the quadratic equation;

`x={-(-4)+-sqrt((-4)^2-4(4)(1))}/(2*4)`

`x={4+-sqrt(16-16)}/8`

`x=(4+-0)/8`

`x=4/8`

`x=1/2`

Hence the value of x is 1/2.

Hence Solved!

4x^2 = 4x - 1

4x^2 - 4x + 1 = 0

Since the above is in the form of quadratic equation i.e. ax^2+bx+c=0, therefore we can use the quadratic formula to find the roots of the equation.

`x = [-b+-sqrt(b^2-4ac)] / (2a)`

Where,

a= 4

b= -4

c= 1

Now input he values in the formula to find the roots,

`x = [-(-4)+-sqrt(-4^2-4*4*1)] / (2*4)`

`x= [4+-sqrt(16-16)] / 8`

`x= [4+-0]/8`

`x= (4+0) / 8 => 1/2`

`x= (4-0)/8 => 1/2`

`4x^2 = 4x - 1`

combine like terms

`4x^2-4x + 1`

a=4 b=-4 c=1

multiply a by c

`4xx1=4 `

find factors of 4 that add up to b which is -4

the numbers will be -2, and -2

now make those numbers b

`4x^2-2x-2x + 1 `

group the numbers

`(4x^2-2x)(-2x + 1)`

factor out common factors

2x(2x-1) -1(2x-1)

`(2x-1)(2x-1)`

set it up to 0

2x-1=0

+1 +1

`2x=1`

`(2x)/2 = 1/2`

`x=1/2`