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. The most typical application of completing the square is to solve a quadratic problem. 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. Completing the square approach is beneficial in the following situations:
- Converting a quadratic expression to its vertex form.
- Computing the minimum/maximum value of the quadratic formula
- A quadratic function is graphed.
- A quadratic equation must be solved.
- The quadratic formula is derived.
What is Completing the Square Method?
The most common application of the completing the square approach is factoring a quadratic equation and thereby determining the roots and zeros of a quadratic polynomial or a quadratic equation. The factorization approach can be used to solve a quadratic equation of the type ax2 + bx + c = 0. However, factoring the quadratic formula ax2 + bx + c is sometimes difficult or impossible.
For example in the case:
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 write it as a(x + m)2 + n. We call this “completing the square” because we have (x + m) fully squared.
Formula for Completing the Square
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 0, can be turned into a perfect square using one additional constant.
Completing the square formula is a methodology or procedure 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.
ax2 + bx + c
where a(x + m)2 + n is the formula for completing the square.
where n seems to be a constant and m could be any real number.
Instead of a lengthy step-by-step approach, we can use the following simple formula to build the square. Find the following to complete the square in the phrase ax2 + bx + c:
n = c – (b2/4a) and m = b/2a
Values substituted in ax2 + bx + c = a(x + m)2 + n. These formulas are geometrically developed.
Steps to Completing the square method
Lets assume the quadratic equation is as ax2 + bx + c = 0. Follow the steps to solve it using the completing the square approach.
Step 1: Form the equation in such a way that c is on the right side.
Step 2 : Divide the entire equation by an if an 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, (b/2a)2.
Step 4 : Factor the left side of the equation as the binomial term’s square.
Step 5: On both sides, take the square root.
Step 6 : Find the roots by solving for variable x.
The steps outlined above can be carried out as illustrated below.
Take a look at the quadratic equation ax2 + bx + c = 0 ( a not equal to 0 ).
By dividing everything by a, we get
x2 + (b/a)x + (c/a) = 0
This can alternatively be written as (b/2a)2 ( by adding and subtracting )
[x + (b/2a)]2 – (b/2a)2 + (c/a) = 0
[x + (b/2a)]2 – [(b2 – 4ac)/4a2] = 0
[x + (b/2a)]2 = [(b2 – 4ac)/4a2]
If b2 – 4ac ≥ 0, then taking the square root, we gets
x + (b/2a) = ± √(b2 – 4ac)/ 2a
The quadratic formula is obtained by simplifying this further.
Sample Questions
Question 1: Find the roots of the quadratic equation of the x2 + 2x – 12 = 0 by using the method of completing the square.
Answer:
Given Quadratic equation is x2 + 2x – 12 = 0
So as comparing the equation along with standard form,
where b = 2, and c = -12
then (x + b/2)2 = -(c – b2/4)
by substituting the values we get
(x + 2/2)2 = -(-12 – (22/4) )
(x + 1)2 = 12 + 1
(x + 1)2 = 13
x + 1 = ± √13
x + 1 = ± 3.6
So, x + 1 = +3.6 and x+1 = – 3.6
x = 2.6 , -4.6
Therefore roots for the given equation are 2.6, -4.6.
Question 2: Find the roots of the quadratic equation of the 2x2 – 4x – 20 = 0 by using the method of completing the square.
Answer:
Given Quadratic equation is 2x2 – 4x – 20 = 0
The supplied equation is not in the form to which the method of completing squares is used, i.e. the x2 coefficient is not 1. To make it one, divide the entire equation by 2 .
then x2 – 2x – 10 = 0
So as comparing the equation along with standard form,
where b = – 2, and c = -10
then (x + b/2)2 = -(c – b2/4)
by substituting the values we get
(x + (-2/2) )2 = -( -10 – (22/4) )
(x – 1)2 = 11
x – 1 = ± √11
x – 1 = ± 3.3
So, x – 1 = + 3.3 and x -1 = -3.3
x = 4.3, -2.3
Therefore roots for the given equation are 4.3, -2.3.
Question 3: Solve Using the completing the square formula for 3x2 – 9x – 27 = 0.
Answer:
Given Quadratic equation is 3x2 – 9x – 27 = 0.
we can write it as x2 – 3x -9 =0
So as comparing the equation along with standard form,
where b = – 3, and c = -9
then (x + b/2)2 = -(c – b2/4)
by substituting the values we get
(x + (-3/2) )2 = -( -9 – (32/4) )
(x – 1.5 )2 = 11.25
x – 1.5 = ± √11.25
x – 1.5 = ± 3.35
So, x – 1.5 = + 11.25 and x -1 = -11.25
x = 12.75, -10.25
Therefore roots for the given equation are 12.75, -10.25.
Question 4: Find the number that needs be added to x2 – 4x to make it a perfect square trinomial using the completing the square formula.
Answer:
Given expression is x2 -4x
As Comparing the given expression along with ax2 + bx + c,
a = 1 ; b = -4
The term that should be added to make the above expression a perfect square trinomial using the formula is,
(b/2a)2 = ( -4/2(1) )2
(b/2a)2 = 4
Therefore the number that needs be added to x2 – 4x to make it a perfect square trinomial is 4.
Question 5: Find the number that needs be added to x2 + 22x to make it a perfect square trinomial using the completing the square formula.
Answer:
Given expression is x2 + 22x
As Comparing the given expression along with ax2 + bx + c,
a = 1 ; b = 22
The term that should be added to make the above expression a perfect square trinomial using the formula is,
(b/2a)2 = ( 22/2(1) )2
(b/2a)2 = 121
Therefore the number that needs to be added to x2 + 22x to make it a perfect square trinomial is 121.
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