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
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
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.
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