Question

How do you solve \[{x^2} + 14x - 15 = 0\]by completing the square?

Answer 1

Hint: Completing the square, or complete the square, is a method that can be used to solve quadratic equation, generally it’s the process of putting an equation of the form \[a{x^2} + bx + c = 0\]in to the form \[{\left( {x + k} \right)^2} + A = 0\] where x is the variable and a, b, c, k and A are constants. Then by the resultant equation on simplification we get the required root or value of variable x.

Complete step-by-step answer:
Completing the square method is one of the methods to find the roots of the given quadratic equation. A polynomial equation with degree equal to two is known as a quadratic equation. ‘Quad’ means four but ‘Quadratic’ means ‘to make square’. A quadratic equation in its standard form is represented as:
 \[a{x^2} + bx + c = 0\], where a, b and c are real numbers such that \[a \ne 0\]and x is a variable
Since the degree of the equation \[a{x^2} + bx + c = 0\] is two; it will have two roots or solutions. The roots of polynomials are the values of x which satisfy the equation. There are several methods to find the roots of a quadratic equation. One of them is by completing the square.
Consider the given equation
  \[ \Rightarrow \,\,{x^2} + 14x - 15 = 0\]
Transform the equation so that the constant term -15, is alone on the right side
 \[ \Rightarrow \,\,{x^2} + 14x = 15\]
Add 49 on both sides, to make the perfect square on the left hand side.
 \[ \Rightarrow \,\,{x^2} + 14x + 49 = 15 + 49\]
49 is the square of 7 i.e., \[{7^2} = 49\]
 \[ \Rightarrow \,\,{x^2} + 14x + {7^2} = 64\]
using the algebraic identity \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\], then we get
  \[ \Rightarrow \,\,{\left( {x + 7} \right)^2} = 64\]
Take square root on both the side, them
 \[ \Rightarrow \,\,x + 7 = \pm \,\sqrt {64} \]
The square root of 64 is 8 i.e., \[\sqrt {64} = \sqrt {{8^2}} = 8\]
 \[ \Rightarrow \,\,x + 7 = \pm \,8\]
 \[ \Rightarrow \,\,x + 7 = 8\] or \[x + 7 = - 8\]
 \[ \Rightarrow \,\,x = 8 - 7\] or \[x = - 8 - 7\]
 \[\therefore \,\,\,\,\,x = 1\] or \[x = - 15\]
Hence, the roots of the equation \[{x^2} + 14x - 15 = 0\]by completing the square method is \[x = 1\] or \[x = - 15\]
So, the correct answer is “\[x = 1\] or \[x = - 15\]”.

Note: The equation is a quadratic equation. Like the perfect square number there is a perfect square equation also. By using the simple arithmetic operations to the equation and applying the square and square root we are going to find the required solution. The perfect squares equations are \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\] and \[{\left( {a - b} \right)^2} = {a^2} - 2ab + {b^2}\]