how to factor 4x^2+4x-1
and x^2-2x+1=5
To factor the quadratic expression 4x^2 + 4x - 1, we look for two binomials in the form (ax + b)(cx + d) that multiply to give us the expression.
1. Start by multiplying the coefficient of the x^2 term (4) with the constant term (-1) to get -4.
In this case, we are looking for two numbers that multiply to -4 and add up to the coefficient of the x-term (4).
The numbers that satisfy this condition are 1 and -4.
2. Rewrite the middle term (4x) using the two numbers from step 1.
4x can be rewritten as 1x - 4x.
3. The quadratic expression can now be written as:
4x^2 + 1x - 4x - 1
4. Group the terms in pairs and factor out the greatest common factor from each pair:
x(4x + 1) - 1(4x + 1)
5. Notice that we now have a common binomial factor of (4x + 1) in both terms.
Factor out (4x + 1) from each term:
(4x + 1)(x - 1)
Therefore, the factored form of 4x^2 + 4x - 1 is (4x + 1)(x - 1).
To factor the quadratic equation x^2 - 2x + 1 = 5, we want to rewrite it in the standard form (ax^2 + bx + c = 0) before factoring.
1. Subtract 5 from both sides of the equation:
x^2 - 2x + 1 - 5 = 5 - 5
2. Simplify the expression:
x^2 - 2x - 4 = 0
Now, we proceed with factoring:
3. We need to find two numbers that multiply to give -4 (coefficient of x^2 term) and add up to -2 (coefficient of x-term).
The numbers that satisfy this condition are -4 and 1.
4. Rewrite the middle term (-2x) using the two numbers from step 3:
-2x can be rewritten as -4x + 2x.
5. The quadratic equation can now be written as:
x^2 - 4x + 2x - 4
6. Group the terms in pairs and factor out the greatest common factor from each pair:
x(x - 4) + 2(x - 2)
7. Notice that we now have a common binomial factor of (x - 2) in both terms.
Factor out (x - 2) from each term:
(x - 2)(x + 2)
Therefore, the factored form of x^2 - 2x + 1 = 5 is (x - 2)(x + 2).