You might be knowing the quadratic equations. Any equation in the form of ax2+bx+c = 0 is termed as quadratic. Where a, b, and c are real numbers such that a ≠ 0 and x is a variable. As it is a second-degree equation, it has got two roots. There are many ways to find these roots. Method of completing the square is the most commonly used.
The method of completing the squares is beneficial for solving, deriving, and graphing quadratic equations. You can also use it in evaluating integrals in calculus and Laplace transforms. Let us learn in detail about this method of completing the squares.
Formula of Completing the Square Method
Solving quadratic equations that cannot be factored in involves completing the square. In this method, the given quadratic equation is manipulated to get the perfect square trinomial on the left side of the equation.
The formula is derived by manipulating the quadratic equation of the form ax2 + bx + c = 0 as given below.
- To the right of the equation, isolate the term c
ax2 + bx = -c
- Divide each term by the coefficient of x2. That is a.
x2 + bx/a = -c/a
- Add the square of half of the coefficient of x, (b/2a)2, on both sides.
x 2 + bx/a + (b/2a)2 = – c/a + (b/2a)2
- Write the LHS term as a perfect square.
(x + b/2a)2= (-4ac+b2)/4a2
- Taking square root on both sides
(x + b/2a) = ±√ (-4ac+b2)/2a
x = – b/2a ±√ (b2– 4ac)/2a
x =(-b±√(b^2-4ac))/2a………. (This is the required quadratic formula)
Steps to Solve a Quadratic Equation using Completing the Square Method
To solve the quadratic equation of the standard form ax2+bx+c = 0 by the method of completing squares follow the below steps.
- Write the given equation in the standard form. Then take c to the RHS side of the equation.
- Make the coefficient of x2 equal to 1 by dividing the entire equation by the given coefficient.
- Add the square of half of the coefficient of x, (b/2a)2, on both sides.
- Write the LHS term as a perfect square.
- Find the square root of the two sides
- Calculate the roots of the variable x.
Let me explain these steps in detail with this example
Example: Find the roots of the quadratic equation 2x2 + 5x – 3 = 0 by the method of completing the square.
Solution: Given: quadratic equation is 2x2 + 5x – 3 = 0. It is in the standard form.
Take c term to the RHS side of the equation
2x2 + 5x = 3
Make the coefficient of x2 equal to 1 by dividing the entire equation by the given coefficient.
2x2 + 5x = 3 2 on both sides
x2 + 5/2x = 3/2
Add the square of half of the coefficient of x, (b/2a)2, on both sides. Here b = 5 and a = 2
x2 + 5/2x + (5/4)2 = 3/2 + (5/4)2
The LHS is in the form of (a + b)2. Hence x2 + 5/2x + (5/4)2 = (x + 5/4)2
(x + 5/4)2 = 3/2 + (25/16) = (24 + 25)/16 = 49/16
Find the square root of the two sides
(x + 5/4)2= 49/16
x + 5/4 = 7/4
x = 7/4 – 5/4 = 2/4 = 1/2
Hence the root of x is ½.
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