Question

How do you solve $y = {x^2} - 12x + 40$ by completing the square?

Answer 1

Hint: The question deals with the concept of quadratic equations. A quadratic equation can be solved by completing the square method. In complete square method we solve the quadratic equation with the use of identity ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$. We make the coefficient of ${x^2}$ equal to one by dividing it with suitable variables. We change the quadratic equation in complete square form so that we can apply the basic identity of algebra. We add or subtract the square of half value of coefficient of $x$ inside the equation having the rest of the term unchanged. Use the identity to write the equation in brackets. We take out the constant value and add or subtract that value with another constant value. Finally we write the quadratic equation in complete square form and the rest of the term keeps unchanged.

Complete step by step solution:
Step: 1 the given quadratic equation is,
${x^2} - 12x + 40$
Take the constant term on the left hand side of the equation and write equal to zero.
${x^2} - 12x = - 40$
The coefficient of ${x^2}$ is one in the given quadratic equation.
Step: 2 consider the value of the coefficient of $x$ in the equation. Half the value of the coefficient of $x$. Square the value of the coefficient of $x$.
The value of the coefficient of $x$is $\left( { - 12} \right)$ and half of the $\left( { - 12} \right)$ is $\left( { - 6} \right)$. The square of the $\left( { - 6} \right)$ is $\left( {36} \right)$. Add $36$ to the equation.
$
   \Rightarrow {x^2} - 12x + 36 = - 40 + 36 \\
   \Rightarrow {x^2} - 12x + 36 = - 4 \\
 $
Step: 3 use the identity formula ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$to change the equation into complete square form.
$
   \Rightarrow {x^2} - 12x + 36 = - 4 \\
   \Rightarrow {\left( {x - 6} \right)^2} = - 4 \\
   \Rightarrow {\left( {x - 6} \right)^2} = 4{i^2} \\
 $
Take the square root to the both sides of the equation.
$
   \Rightarrow \left( {x - 6} \right) = \sqrt {4{i^2}} \\
   \Rightarrow x - 6 = \pm 2i \\
 $
Solve the equation by taking constant terms on one side of the equation.
$x = 6 \pm 2i$

Final Answer:
Therefore the answer of the equation by complete square method is $x = 6 \pm 2i$.

Note:
Make the coefficient of the ${x^2}$ one. Consider the value of coefficient of $x$ and, take half of the value of coefficient of $x$, square the half value then add it to the both sides of the equation. Use the basic algebra identity ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$ to convert the equation in complete square form.