Question

How do you solve $ {x^2} - 10x + 25 = 0 $ ?

Answer 1

Hint: We are given a quadratic equation to solve. We can solve a quadratic equation by factoring method. In this method we have to split the middle term i.e., the coefficient of $ x $ in such a way that the product of the two terms in which it got split should be equal to the product of the leading coefficient and the constant. After this splitting, we will take the common factor out from the first two terms and the last two terms and then from the whole expression. This is how the quadratic will be factorized and we will get its solutions.

Complete step-by-step answer:
(i)
We are given:
 $ {x^2} - 10x + 25 = 0 $
As we know that it is a quadratic equation, so in order to solve it, we will apply the factoring method. Thus, we have to split the middle term i.e., $ - 10 $ in such a way that the product of the two terms in which it got split should be equal to the product of the leading coefficient i.e., $ 1 $ and the constant $ 25 $ .
As we know that the sum of $ - 5 $ and $ - 5 $ is $ - 10 $ and their product is $ 25 $ , we can split $ - 10 $ as the sum of $ - 5 $ and $ - 5 $ . Therefore, we will get:
 $ {x^2} - 5x - 5x + 25 = 0 $
(ii)
Now we will take the common factor out from the first two terms and the last two terms. Therefore, we will get:
 $ x\left( {x - 5} \right) - 5\left( {x - 5} \right) = 0 $
Now we will take the common factor $ \left( {x - 5} \right) $ out from the whole expression. Therefore, we will get:
 $ \left( {x - 5} \right)\left( {x - 5} \right) = 0 $
(iii)
Now since, we have got our quadratic equation in the form of a product of two factors, we will directly equate both the factors with zero to obtain the roots of the equation. Therefore,
 $ x - 5 = 0 $
And,
 $ x - 5 = 0 $
As we have both the factors the same, we know that we will get the same value of $ x $ from both the factors. Therefore, adding $ 5 $ on both the sides of the factor, we will get:
 $
  x - 5 + 5 = 0 + 5 \\
  x = 5 \\
  $
Therefore, at $ x = 5 $ , the given equation becomes $ 0 $ .
Since, we know that it is a quadratic equation and that a quadratic equation always has two roots, we will say that the quadratic equation $ {x^2} - 10x + 25 = 0 $ has two equal roots that are $ 5 $ and $ 5 $ .
So, the correct answer is “5 AND 5”.

Note: If we face difficulty while splitting the middle term and finding factors, we can directly use the quadratic formula $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ in order to solve the quadratic equation given to us. This formula is always applicable and will give the same result as the factoring method. Also, we can verify our answer by checking the nature of the roots with the help of discriminant $ D $ . If $ D > 0 $ , we will have two separate real roots. If $ D < 0 $ , we will have two imaginary roots. And if $ D = 0 $ , we will have two equal roots for the given quadratic equation where $ D = {b^2} - 4ac $ and the equal roots will be $ x = \dfrac{{ - b}}{{2a}} $ .