Question

How do you solve ${x^2} - 4x = 10$ by completing the square?

Answer 1

Hint: We just compare the given equation to the general form of quadratic equation $a{x^2} + bx + c = 0$ .To solve $a{x^2} + bx + c = 0$ by completing the square method, halve the variable of x, square the resultant and add the number to both the sides of the equation.

Complete step-by-step solution:
Given a quadratic equation and we have been asked to solve the quadratic equation by completing the square method.
The given quadratic equation as below:
 ${x^2} - 4x = 10$
Here $a = 1,b = - 4$
Now, we need to find half of the given coefficient of $x$ and then, find square of half of $b.$
${\left( {\dfrac{b}{2}} \right)^2} = {\left( {\dfrac{{ - 4}}{2}} \right)^2} = {( - 2)^2}$
Add the same term to each side of the equation.
We can have,
$\Rightarrow$${x^2} - 4x + {( - 2)^2} = 10 + {( - 2)^2}$
Opening the squares, we get,
$\Rightarrow$${x^2} - 4x + 4 = 10 + 4$
Adding the terms on the right-hand side, we get,
$\Rightarrow$${x^2} - 4x + 4 = 14$
We know that ${a^2} - 2ab + {b^2} = {(a - b)^2}$
$\Rightarrow$${x^2} - 2.x.2 + {2^2} = 14$
Here $a = x,b = 2$
Hence solving the given equation according to the completing the square method as given below
Factor the perfect trinomial square into ${(x - 2)^2}$.
$\Rightarrow$${(x - 2)^2} = 14$
Squaring rooting both sides, we get,
$\Rightarrow$$(x - 2) = \pm \sqrt {14} $
Taking $ - 2$ to the right-hand side and add to the right-hand side term,
We get,
$\Rightarrow$$x = \pm \sqrt {14} + 2$
Solve the equation of $x.$
$\Rightarrow$$x = \sqrt {14} + 2$ and $x = - \sqrt {14} + 2$
Decimal form:
$\Rightarrow$$x = 5.7465738...., - 1.74165738....$

So, $x = \sqrt {14} + 2$ and $x = - \sqrt {14} + 2$ is the solution for the required question.

Note: This type of problem deals with solving the given quadratic equation with the completing the square method. Completing the square is a method used to solve a quadratic equation by changing the form of the equation so that the left side is a perfect square trinomial.
While solving this problem please be careful when considering the terms, which of the one will be $a$ , and which one will be $b.$
There are other methods to solve a quadratic equation. They are:
Splitting the middle term method
Quadratic equation formula
You can use any method if nothing is mentioned. But if they have mentioned the method, you have to use that method only.