Question

How do you solve the quadratic equation by completing the square: $$x^2-6x=0$$

Answer 1

Hint: This question belongs to the topic quadratic equation. In this question, first we will add the square of half coefficient of x to the both sides of the equation. After that, we will make the equation in the form of $$a^2-2ab+b^2$$. As we know that the equation $$a^2-2ab+b^2$$ is the perfect square of $$(a-b)$$. So, we will use this formula and solve the further solution to get the value of x.

Complete step by step answer:
Let us solve this question.
In this question, we have to solve the quadratic equation $$x^2-6x=0$$ by completing the square.
For solving this question by completing the square, firstly we will make sure that the coefficient of $$x^2$$ is 1. We can see in the equation $$x^2-6x=0$$ that coefficient of $$x^2$$ is 1. Now, we can solve further.
In the given equation, we are going to add the square of half of coefficient of x (that is 9) to the both of equation. We will get,
$$x^2-6x+9=0+9$$
The above equation can also be written as
$$x^2-2\times 3\times x+3^2=3^2$$
As we can see that left hand side of equation is in the form of $$a^2-2ab+b^2$$ and we know that the equation $$a^2-2ab+b^2$$ is equal to $${(a-b)}^2$$. By comparing both the equations, we can say that a=x and b=3. So, we can write the above equation as
$$\Rightarrow {(x-3)}^2=3^2$$
Now, taking the square root to both the side of equation, we get
$$\Rightarrow \sqrt{{(x-3)}^2}=\pm \sqrt{3^2}$$
We can write the above equation as
$$\Rightarrow (x-3)=\pm 3$$
We can write the equation as
 $$\Rightarrow x=3\pm 3$$
From the above equation, we have got the two values of x.
x=3-3 and x=3+3

From the above, we can say that the values of x are 0 and 6.

Note: We should have a better knowledge in the topic quadratic equation. Don’t forget to give both plus and minus signs while taking the square root of a number. Remember the following formula to solve this type of question easily:
$$a^2-2ab+b^2={(a-b)}^2$$
 We have a different method to solve this question.
As we have found from the above that
$${(x-3)}^2=3^2$$
We can solve from here by a different method.
The above equation can also be written as
$$\Rightarrow {(x-3)}^2-3^2=0$$
As we know that $$a^2-b^2=(a-b)(a+b)$$. Using this formula in the above by putting a=x-3 and b=3, we can write
$$\Rightarrow (x-3-3)(x-3+3)=0$$
The above equation can also be written as
$$\Rightarrow (x-6)(x+0)=0$$
From here, we can say that
x=-6 and x=0
We get the same solution. So, we can use this method too.