COMPLETING THE SQUARE

A quadratic equation is a polynomial equation with a degree equal to two. The word ‘quad’ signifies four, yet the word ‘quadratic’ means ‘to square.’ In its simplest form, a quadratic equation is written as: ax²+ bx + c = 0 , where a , b , and c are real numbers and a is not equal to 0 and x is variable.

Because the above-written equation has a degree of two, it will have two roots or solutions. The values of x that fulfil the equation are the roots of polynomials. Finding the roots of a quadratic equation can be done in a number of ways. Completing the square is one of the ways to solve the quadratic equation.

COMPLETING THE SQUARE:-

One of the approaches for finding the roots of a given quadratic equation is to complete the square method. We must turn the provided equation into a perfect square using this method.

A method for turning a quadratic formula of the form ax² + bx + c to the vertex form a(x – h)² + k is known as completing the square. Solving a quadratic equation is the most common use of completing the square. This can be accomplished by rearranging the expression a(x + m)² + n obtained after completing the square, so that the left side is a perfect square trinomial. The completing square method is useful in 

• Taking a quadratic expression and converting it to vertex form.
• Identifying the least and maximum values of a quadratic expression.
• A quadratic function is graphed.
• Using a calculator to solve a quadratic problem.
• The quadratic formula is derived in this way.

COMPLETING THE SQUARE METHOD:-

Factoring a quadratic equation, and thus finding the roots and zeros of a quadratic polynomial or a quadratic equation, is the most typical use of the completing the square method. The factorization approach can be used to solve a quadratic equation of the type ax² + bx + c = 0. However, factoring the quadratic expression ax² + bx + c can be difficult or impossible at times.

Example:- We can’t factorise x² + 2x + 3 because we can’t find two numbers whose sum is 2 and whose product is 3. In such circumstances, we complete the square and express it as a(x + m)² + n. We claim we’ve “finished the square” here because we have (x + m) full squared.

COMPLETING THE SQUARE FORMULA:-

A methodology or approach for converting a quadratic polynomial or equation into a perfect square with an additional constant is known as the square formula. Using the completing the square formula or approach, a quadratic expression in variable x: ax² + bx + c, where a, b, and c are any real values except a which can not be zero , can be turned into a perfect square using one additional constant.

The quadratic formula is derived by completing the square formula.

Completing the square formula is a methodology or approach for finding the roots of specified quadratic equations, such as ax² + bx + c = 0, where a, b, and c are all real values except a which can not be zero.

FORMULA FOR COMPLETING SQUARE:-

ax² + bx + c = a(x + m)² + n is the formula for completing the square.

where n is a constant term and m is any real number.

Instead of employing a complicated step-by-step procedure to construct the square, we can utilise the following simple formula. Find the following to complete the square in the phrase ax2 + bx + c:

Finding m = b ⁄ 2a      and n= c-b²⁄ 4a

Putting these values in the equation ax² + bx + c = a(x + m)² + n and then we will get the required value of zeros easily.

STEPS FOR COMPLETING THE SQUARE:-

Assume the quadratic equation is ax² + bx + c = 0. Then, using the completing the square method, follow the steps to solve it.

Step 1:- Form the equation in such a way that c is on the right side.

Step 2:- Divide the entire equation by a if a is not equal to 1, such that the coefficient of x2 equals 1.

Step 3:- On both sides, add the square of half of the term-x coefficient, i.e. (b ⁄ 2a)²

Step 4:- Factor the left side of the equation as the binomial term’s square.

Step 5:- Applying square root on both sides.

Step 6 :- finally solving for x and getting the roots of the equation.

Example 1 :- Find the roots of the equation X²+5x + 6 = 0 using the completing square method.

Solution:- Here we have the quadratic equation

X² + 5x +6 =0 and we get a=1 , b = 5 and c = 6

• (x + b/2)² = -(c – b²/4)

So, [x + (5/2)]² = -[6 – (25/4)]

=> (x + 5/2)² = 1/4

=> (x + 5/2)² = 1/4

=> (x + 5/2) = ±√1/4

=> (x + 5/2) = ± 1/2

=> x + 5/2 = ½ and x + 5/2 = -½

=> x = -2 , -3     

Hence the roots of the equation are 1 and -5.

CONCLUSION:-

Completing the Square is a method for solving a quadratic equation by changing the shape of the equation to a perfect square trinomial on the left side. Completing the square is a method of computing quadratic polynomials in mathematics. ax² + bx + c = (x + p)² + constant is the formula for completing the Square Formula.