# COMPLETING THE SQUARE METHOD CLASS 10

Completing the Square Method Class 10 :

In this section, you will learn how to solve quadratic equation using by completing the square method..

To apply completing the square method, the quadratic equation must be in the form of

ax2 + bx + c  =  0

## Solving Quadratic Equations by Completing the Square - Steps

Step 1 :

In the given quadratic equation ax2 + bx + c = 0, divide the complete equation by a (coefficient of x2).

If the coefficient of x2 is 1 (a = 1), the above process is not required.

Step 2 :

Move the number term (constant) to the right side of the equation.

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2.

Examples :

6x should be written as 2(3)(x).

5x should be written as 2(x)(5/2).

Step 4 :

The result of step 3 will be in the form of

x2 + 2(x)y  =  k

Step 4 :

Now add y2 to each side to complete the square on the left side of the equation.

Then,

x2 + 2(x)y + y2  =  k + y2

Step 5 :

In the result of step 4, if we use the algebraic identity

(a + b)2  =  a2 + 2ab + b2

on the left side of the equation, we get

(x + y)2  =  k + y2

Step 6 :

Solve (x + y)2  =  k + yfor x by taking square root on both sides.

## Solving Quadratic Equations by Completing the Square Method - Examples

Example 1 :

Solve the following quadratic equation by completing the square method.

x2 + 6x - 7  =  0

Solution :

Step 1 :

In the quadratic equation x2 + 6x - 7 = 0, the coefficient of x2 is 1.

So, we have nothing to do in this step.

Step 2 :

Add 7 to each side of the equation x2 - 6x - 7  =  0.

x2 - 6x  =  7

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2.

Then,

x2 - 6x  =  7

x2 - 2(x)(3)  =  7

Step 4 :

Now add 32 to each side to complete the square on the left side of the equation.

Then,

x2 - 2(x)(3) + 32  =  7 + 32

(x - 3)2  =  7 + 9

(x - 3)2  =  16

Take square root on both sides.

(x - 3)2  =  √16

x - 3  =  ±4

x - 3  =  -4  or  x - 3  =  4

x  =  -1  or  x  =  7

So, the solution is {-1, 7}.

Example 2 :

Solve the following quadratic equation by completing the square method.

x2 + 3x + 1  =  0

Solution :

Step 1 :

In the quadratic equation x2 + 3x + 1 = 0, the coefficient of x2 is 1.

So, we have nothing to do in this step.

Step 2 :

Subtract 1 from each side of the equation x2 + 3x + 1 = 0.

x2 + 3x  =  -1

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2.

Then,

x2 + 3x  =  -1

x2 + 2(x)(3/2)  =  -1

Step 4 :

Now add (3/2)2 to each side to complete the square on the left side of the equation.

Then,

x2 + 2(x)(3/2) + (3/2)2  =  -1 + (3/2)2

(x + 3/2)2  =  -1 + 9/4

(x + 3/2)2  =  -4/4 + 9/4

(x + 3/2)2  =  (-4 + 9)/4

(x + 3/2)2  =  5/4

Take square root on both sides.

(x + 3/2)2  =  √(5/4)

x + 3/2  =  ±√5/2

x + 3/2  =  -√5/2  or  x + 3/2  =  √5/2

x  =  -√5/2 -3/2  or  x  =  5/2 - 3/2

x  =  (-√5 - 3)/2  or  x  =  (√5 - 3)/2

x  =  -(√5 + 3)/2  or  x  =  (√5 - 3)/2

So, the solution is {-(√5 + 3)/2, (√5 - 3)/2}.

Example 3 :

Solve the following quadratic equation by completing the square method.

2x2 + 5x - 3  =  0

Solution :

Step 1 :

In the given quadratic equation 2x2 + 5x - 3 = 0, divide the complete equation by 2 (coefficient of x2).

x2 + (5/2)x - (3/2)  =  0

Step 2 :

x2 + (5/2)x  =  3/2

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2.

Then,

x2 + 2(x)(5/4)  =  3/2

Step 4 :

Now add (5/4)2 to each side to complete the square on the left side of the equation.

Then,

x2 + 2(x)(5/4) + (5/4)2  =  3/2 + (5/4)2

(x + 5/4)2  =  3/2 + 25/16

(x + 5/4)2  =  24/16 + 25/16

(x + 5/4)2  =  (24 + 25)/16

(x + 5/4)2  =  49/16

Take square root on both sides.

√(x + 5/4)2  =  √(49/16)

x + 5/4  =  ± 7/4

x + 5/4  =  -7/4  or  x + 5/4  =  7/4

x  =  -7/4 - 5/4  or  x  =  7/4 - 5/4

x  =  (-7 - 5)/4  or  x  =  (7 - 5)/4

x  =  -12/4  or  x  =  2/4

x  =  -3  or  x  =  1/2

So, the solution is {-3, 1/2}.

Example 4 :

Solve the following quadratic equation by completing the square method.

4x2 + 4bx - (a2 - b2)  =  0

Solution :

Solution :

Step 1 :

In the given quadratic equation 4x2 + 4bx - (a2 - b2)  =  0, divide the complete equation by 4 (coefficient of x2).

x2 + bx - (a2 - b2)/4  =  0

Step 2 :

Add (a2 - b2)/4 to each side.

x2 + bx  =  (a2 - b2)/4

Step 3 :

In the result of step 2, write the "x" term as a multiple of 2.

Then,

x2 + bx  =  (a2 - b2)/4

x2 + 2(x)(b/2)  =  (a2 - b2)/4

Step 4 :

Now add (b/2)2 to each side to complete the square on the left side of the equation.

Then,

x2 + 2(x)(b/2) + (b/2)2  =  (a2 - b2)/4 + (b/2)2

(x + b/2)2  =  (a2 - b2)/4 + b2/4

(x + b/2)2  =  (a2 - b+ b2)/4

(x + b/2)2  =  a2/4

Take square root on both sides.

√(x + b/2)2  =  √(a2/4)

x + b/2  =  ± a/2

x + b/2  =  -a/2  or  x + b/2  =  a/2

x  =  -a/2 - b/2  or  x  =  a/2 - b/2

x  =  (-a - b)/2  or  x  =  (a - b)/2

x  =  -(a + b)/2 or  x  =  (a - b)/2

So, the solution is {-(a + b)/2, (a - b)/2}. After having gone through the stuff given above, we hope that the students would have understood how to solve quadratic equations by completing the square method.

Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

You can also visit our following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6 