Question

How do you make ${x^2} + 10x + c$ a perfect square?

Answer 1

Hint: The question deals with the concept of quadratic equations. To make a perfect square we have to find the value of the constant $c$. 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 the 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 keeping unchanged.

Complete step by step solution:
Step: 1 the given quadratic equation is,
${x^2} + 10x + c$
 We know that the form of a perfect square that is,
${\left( {x + m} \right)^2} = {x^2} + 2mx + {y^2}$
 Step: 2 now we will assume that the given quadratic equation is a perfect square and then we will compare it with the given form of the perfect square equation.
$ \Rightarrow {\left( {x + m} \right)^2} = {x^2} + 2mx + {m^2}$
And,
$ \Rightarrow {x^2} + 10x + c$
Here by comparison the value of $c$ is equal to ${m^2}$.
$ \Rightarrow c = {m^2}$
Now compare the coefficient of $x$ in the given quadratic equation with the perfect square equation,
$
   \Rightarrow 2m = 10 \\
   \Rightarrow m = 5 \\
 $
Now find the value of $c$ to make the given quadratic equation a perfect square.
Since $c = {m^2}$
Therefore,
$ \Rightarrow c = 25$
Now substitute the value of $c$ in the given quadratic equation to make it perfect square.
$ \Rightarrow {x^2} + 10x + 25 = {\left( {x + 5} \right)^2}$

Final Answer:
Therefore at the value of $c = 25$ , the given quadratic equation is a perfect square.

Note:
Use the basic algebra identity ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$ to convert the equation in complete square form. Find the value of the constant term in the given quadratic equation to make it a perfect square. Students are advised to compare the given quadratic equation with a perfect square equation.