**Completing the Square Method Examples with Answers :**

In this section, we will learn how to solve quadratic equations using completing the square method.

**Step 1 :**

In the given quadratic equation ax^{2} + bx + c = 0, divide the complete equation by a (coefficient of x^{2}).

If the coefficient of x^{2} 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

x^{2} + 2(x)y = k

**Step 4 :**

Now add y^{2} to each side to complete the square on the left side of the equation.

Then,

x^{2} + 2(x)y + y^{2} = k + y^{2}

**Step 5 :**

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

(a + b)^{2} = a^{2} + 2ab + b^{2}

on the left side of the equation, we get

(x + y)^{2} = k + y^{2}

**Step 6 :**

Solve (x + y)^{2} = k + y^{2 }for x by taking square root on both sides.

**Example 1 :**

Solve by completing the square method

x^{2} – (√3 + 1)x + √3 = 0

**Solution :**

x^{2} – (√3 + 1)x + √3 = 0

x^{2} – (√3 + 1)x = -√3

x^{2} – (√3 + 1)x + [(√3 + 1)/2]^{2 }= -√3 + [(√3 + 1)/2]^{2}

[x - (√3 + 1)/2]^{2 }= -√3 + [(√3 + 1)^{2} / 4]

[x - (√3 + 1)/2]^{2 }= (-4√3 + √3^{2} + 2√3 + 1)/4

[x - (√3 + 1)/2]^{2 }= (√3^{2} - 2√3 + 1)/4

[x - (√3 + 1)/2]^{2 }= [ (√3 - 1)/2 ]^{2}

[x - (√3 + 1)/2]^{2 }= [ (√3 - 1)/2 ]^{2}

[x - (√3 + 1)/2]^{ }= ± (√3 - 1)/2

x - (√3+1)/2 = (√3-1)/2 x = (√3-1)/2 + (√3+1)/2 x = 2√3/2 x = √3 |
x - (√3+1)/2 = -(√3-1)/2 x = -(√3-1)/2 + (√3+1)/2 x = 2/2 x = 1 |

Hence the solution is { 1, √3 }.

**Question 6 :**

Solve by completing the square method

(5x + 7)/(x – 1) = 3x + 2

**Solution :**

(5x + 7) = (3x + 2)(x – 1)

5x + 7 = 3x² – 3x + 2x – 2

3x² – 3x + 2x – 2 – 5x – 7 = 0

3x² – 6x – 9 = 0

Divide the entire equation by 3, we get

x² – 2x – 3 = 0

x² – 3x + x – 3 = 0

x (x – 3) + 1 (x – 3) = 0

(x – 3)(x + 1) = 0

x - 3 = 0 x = 3 |
x + 1 = 0 x = -1 |

Hence the solution is { -1, 3 }.

After having gone through the stuff given above, we hope that the students would have understood how to solve quadratic equations using the completing the square.

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

HTML Comment Box is loading comments...

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

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**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**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**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 **

**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 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**