Question

How do you factor the expression $25{{x}^{2}}-20x+4$ ?

Answer 1

Hint: While factor any quadratic equation $a{{x}^{2}}+bx+c$ , we write the equation $a{{x}^{2}}+bx+c$ as $a{{x}^{2}}+fx+gx+c$ such that the product of f and g is equal to product of a and c. In this case we will write -20x as fx + gx such that the product of f and g is equal to 100. Then we can factorize the equation.

Complete step by step solution:
The given equation is $25{{x}^{2}}-20x+4$.
 If we compare this equation to a standard quadratic equation $a{{x}^{2}}+bx+c$, a is equal to 25 , b=- 20 and c is 4
The product of a and c is 100
So to factorize $25{{x}^{2}}-20x+4$ we will write -20x as fx + gx such that the product of f and g is equal to 100.
The value of f and g will be -10 and -10 , because we can see that the sum of -10 and -10 is -10, product of -10 and -10 is 100.
$\Rightarrow 25{{x}^{2}}-20x+4=25{{x}^{2}}-10x-10x+4$
We can take 5x common form the first 2 term of the equation and -2 common from the second 2 term of the equation
$\Rightarrow 25{{x}^{2}}-20x+4=5x\left( 5x-2 \right)-2\left( 5x-2 \right)$
Now we can take 5x – 2 from whole equation
$\Rightarrow 25{{x}^{2}}-20x+4=\left( 5x-2 \right)\left( 5x-2 \right)$

Note: If we observe we can solve the above question by using algebraic formula ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ . If we put a equal to 5x and b equal to 2 in the equation ${{a}^{2}}-2ab+{{b}^{2}}$ , we will get the equation $25{{x}^{2}}-20x+4$ and the factorization form will be ${{\left( a-b \right)}^{2}}$ which is ${{\left( 5x-2 \right)}^{2}}$