Question
Question: How do you factor \[{y^2} - 4y + 4\]?...
How do you factor y2−4y+4?
Solution
A polynomial of degree two is called a quadratic polynomial and its zeros can be found using many methods like factorization, completing the square, graphs, quadratic formula etc. The quadratic formula is used when we fail to find the factors of the equation. If factors are difficult to find then we use Sridhar’s formula to find the roots. That is y=2a−b±b2−4ac.
Complete step-by-step solution:
Given, y2−4y+4
Since the degree of the equation is 2, we have 2 factors.
On comparing the given equation with the standard quadratic equationay2+by+c, we havea=1, b=−4 and c=4.
The standard form of the factorization of quadratic equation is ay2+b1y+b2y+c, which satisfies the condition b1×b2=a×c and b1+b2=b.
We can write the given equation as y2−2y−2y+4, where b1=−2 and b2=−2. Also b1×b2=(−2)×(−2)=4(ac) and b1+b2=−2−2=−4(b).
Thus we have,
⇒y2−4y+4=y2−2y−2y+4
=y2−2y−2y+4
Taking ‘y’ common in the first two terms and taking −2y common in the remaining two terms we have,
⇒y(y−2)−2(y−2)
Again taking (y−2) common we have,
=(y−2)(y−2)
Hence the factors of y2−4y+4 are (y−2) and (y−2)
Additional information:
If we know the algebraic identity (a−b)2=a2−2ab+b2 we can solve this problem directly and easily.
Compared with the given problem that is y2−4y+4. We have a=y and b=2.
⇒y2−4y+4=(y−2)2
⇒(y−2)(y−2). In both cases we have the same answer.
Thus the factors of the given question are =(y−2)(y−2)
Note: We can also find the roots of the given quadratic equation by equating the obtained factors to zero. That is
(y−2)(y−2)=0
By zero product principle we have,
(y−2)=0 or (y−2)=0
⇒y=2 and y=2.
These are the roots of the given polynomial of degree 2. In above, if we are unable to expand the middle term of the given equation into a sum of two numbers then we use a quadratic formula to solve the given problem. Quadratic formula and Sridhar’s formula are both the same. Careful in the calculation part.