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Completing square method : The standard form of quadratic equation is axSteps used in finding the roots by completing square method are as follows:

1) a$x^{2}$ + bx + c = 0 |
2x^{2}+ 9x – 20 = 0 a = 2, b = 9 and c = -20 |

2) Divide by "a" on both sides$x^{2}$ + bx/ a + c/a = 0 |
Divide by "2" on both sides $x^{2}$ + 9x/ 2 -10 = 0 |

3) add - c on both sides $x^{2}$ + bx/a + c/a = 0 - c/a = –c /a ---------------------- $x^{2}$ + bx/a = -c/a |
add + 10 on both sides $x^{2}$ + 9x/2 - 10 = 0 +10 = –20 --------------------- $x^{2}$ + 9x/2 = 10 |

4) add $b^{2}$/ 4$a^{2}$on both sides$x^{2}$ + bx/a + $b^{2}$/ 4$a^{2}$ = - c/a + $b^{2}$/ 4$a^{2}$ |
add $9x^{2}$/4 x $2^{2}$= 81/16 $x^{2}$ + 9x/2 + 81/16 = 10 + 81/ 16 |

5) $(x + b/2a)^{2}= - c/a+ b^{2}/ 4a^{2}$ |
$(x + 9/4)^{2}$ = (160 + 81)/16 |

6) $(x + b/2a)^{2} = (-4ac + b^{2})/ 4a^{2}$ |
$(x + 9/4)^{2}$ = 241/16 |

7)$ (x + b/2a)^{2}=( b^{2} - 4ac)/4a^{2}$ |
$(x + 9/4)^{2} $= 241/16 |

8) Taking square root on both sides(x + b/2a) =$\pm ( \sqrt(b^{2}$ - 4ac )/ 2a |
Taking square root on both sides (x + 9/4) = $\pm$ √241 /4 |

9)Add - b/2a on both sides x = - b/2a ± (√b ^{2} - 4ac )/2a |
Add - 9/4 on both sides x = - 9/4 ± √241 /4 |

10) x = (- b ± √b^{2} - 4ac )/2a |
x = (- 9 ± √241 )/ 4 |

As we know that ax

1) Move 'c' to other side.

2) Make the leading coefficient as 1.

3) Add

1) Find the roots of the equation 9x

9x

9x

Here, leading coefficient = a = 9 , b = -15 and c = 6

Divide the whole equation by 9, we get

x

x

Now add (-b/2a)

x

( x - 5/6)

( x - 5/6)

( x - 5/6)

x - 5/6 = ± 1/6

∴ x = 5/6 ± 1/6

x = 5/6 + 1/6 or x = 5/6 - 1/6

x = 6/6 = 1 or x = 4/6 = 2/3

So, roots of the equation are 1 and 2/3.

________________________________________________________________

2) Find the roots of the equation 2x

2x

2x

Here, leading coefficient = a = 2 , b = -5 and c = 3

Divide the whole equation by 2, we get

x

x

Now add (-b/2a)

x

( x - 5/4)

( x - 5/4)

( x - 5/4)

x - 5/4 = ± 1/4

∴ x = 5/4 ± 1/4

∴ x = 5/4 + 1/4 or x = 5/4 - 1/4

∴ x = 6/4 = 3/2 or x = 4/4 = 1

So roots of the equation is 3/2 and 1.

• Splitting of middle term

• Completing square method

• Factorization using Quadratic Formula

• Solved Problems on Quadratic Equation

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