Question

How do you factor $ {x^2} - 10x + 25 $ ?

Answer 1

Hint: In this question, we need to solve the equation $ {x^2} - 10x + 25 $ . For splitting the middle term into two factors, we will determine the factors that multiply to give $ ac $ i.e., $ 1 \times 25 = 25 $ , and add to give $ b $ i.e., $ - 10 $ which is called sum-product pattern. Then, factor the first two and last two terms separately. If we have done this correctly, then two new terms will have a clearly visible common factor. Finally, we will equate the factors to $ 0 $ and determine the value of $ x $ .

Complete step-by-step answer:
Now, we need to solve $ {x^2} - 10x + 25 $ .
First, let us determine the factors of the given equation.
According to the rule to factorize,
Product= $ {x^2} $ coefficient $ \times $ constant
And, sum= $ x $ coefficient
Thus, we will find two numbers that multiply to give $ ac $ i.e., $ 1 \times 25 = 25 $ and add to give $ b $ i.e., $ - 10 $ ,
Here, the product is positive but $ b $ is negative. So, we can say that both the factors are negative.
Now, let’s consider the possible factors and their sum.
$
- 25 \times - 1 = 25; - 25 + \left( { - 1} \right) = - 26 \\
 - 5 \times - 5 = - 25; - 5 + \left( { - 5} \right) = - 10 \;
$
From this it is clear that the factors are $ - 5 $ and $ - 5 $ .
Now, by rewriting the middle term with those factors, we have,
 $ {x^2} - 5x - 5x + 25 = 0 $
 $ \left( {{x^2} - 5x} \right) - \left( {5x - 25} \right) = 0 $
Factor out the greatest common factor from each group,
 $ x\left( {x - 5} \right) - 5\left( {x - 5} \right) = 0 $
Factor the polynomial by factoring out the greatest common factor, $ x - 5 $ ,
 $ \Rightarrow \left( {x - 5} \right)\left( {x - 5} \right) = 0 $
Hence, the factors are $ \left( {x - 5} \right) $ and $ \left( {x - 5} \right) $ .
So, the correct answer is “ $ \left( {x - 5} \right) $ and $ \left( {x - 5} \right) $ ”.

Note: In this question it is important to note that this factorization method works for all quadratic equations. The standard form of the quadratic equation is $ a{x^2} + bx + c = 0 $ . It is called factoring because we find the factors. A factor is something we multiply by. There is no simple method of factoring a quadratic expression, but with a little practice it becomes easier. If the question is to solve the equation, then we can finally, equate the equation to $ 0 $ which is common in all quadratic equations because we need to determine the value of the given unknown variable.