SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE WHEN A IS NOT 1

There are three ways to solve a quadratic equation.

i) Using factoring

ii) Quadratic formula

iii) Completing the square.

Here we are going to see, how to solve quadratic equation using completing the square method.

If the quadratic equation is in the form of ax2 + bx + c = 0

Step 1 :

Move the constant to the other side of the equal sign.

Step 2 :

Check if the leading coefficient of x2 is 1. If it is not 1, then divide the quadratic equation by the leading coefficient.

Step 3 :

Write the coefficient of x as multiple of 2.

Step 4 :

So far, the equation will be in the form of a2 + 2ab (or) a2 - 2ab

Step 5 :

Add b2 on both sides, and complete the formula for (a+b)2 or (a-b)2

Step 6 :

Using square root property, solve for the variable x.

Solve for exact values of x by completing the square :

Problem 1 :

2x+ 4x + 1 = 0

Solution :

2x2+ 4x + 1 = 0

Divide each side by 2.

x2 + 2x + 12 = 0x2 + 2x = -12x2 + 2(x)(1) = -12

Add 12 on each sides.

x2 + 2(x)(1) + 12 = -12 + 12(x + 1)2 = -12 + 1(x + 1)2 = -1 + 22(x + 1)2 = 12

Take square root on both sides.

(x + 1)2 = 12x + 1 =± 12x = -1 ± 12

Problem 2 :

2x2 - 10x + 3 = 0

Solution :

2x2 - 10x + 3 = 0

Divide each side by 2.

x2 - 5x + 32 = 0x2 - 5x = -32x2 - 2(x)52 = -32

Add (5/2)2 on each sides.

x2 - 2(x)52 + 522 = -32 + 522x - 522 = -32 + 254x - 522 = -6 + 254x - 522 = 194

Take square root on both sides.

x - 522 = 194x - 52 =± 192x = 52 ± 192

Problem 3 :

3x2 + 12x + 5 = 0

Solution :

3x2 + 12x + 5 = 0

Divide each side by 3.

x2 + 4x + 53 = 0x2 + 4x = -53x2 + 2(x)(2) = -53

Add 22 on each sides.

x2 + 2(x)(2) + 22 = -53 + 22(x + 2)2 = -53 + 4(x + 2)2 = -5 + 123(x + 2)2 = 73

Take square root on both sides.

(x + 2)2 = 73x + 2 =± 73x = -2 ± 73

Problem 4 :

3x2 = 6x + 4

Solution :

3x2 = 6x + 4

3x2 - 6x = 4

Divide each side by 3.

x2 - 2x = 43x2 - 2(x)(1) = 43

Add 12 on each sides.

x2 - 2(x)(1) + 12 = 43 + 12(x - 1)2 = 43 + 1(x - 1)2 = 4 + 33(x - 1)2 = 73

Take square root on both sides.

(x - 1)2 = 73x -1= ±73x = 1 ± 73

Problem 5 :

5x2 - 15x + 2 = 0

Solution :

5x2 - 15x + 2 = 0

Divide each side by 5.

x2 - 3x + 25 = 0x2 - 3x = - 25 x2 - (2)(x)32= -25

Add (3/2)2 on each sides.

x2 - (2)(x)32+322= -25 + 322x - 322 = -25 + 94x - 322 = -820 + 4520x - 322 = 3720

Take square root on both sides.

x - 322 = 3720x - 32 = ±3720x = 32 ± 3720

Problem 6 :

4x2 + 4x = 5

Solution :

4x2 + 4x = 5

Divide each side by 4.

x2 + x = 54x2 + 2(x)12 = 54

Add 12 on each sides.

x2 + 2(x)12 + 122 = 54 + 122x + 122 = 54 + 14x + 122 = 5 + 14x + 122 = 64

Take square root on both sides.

x + 122 = 64x + 12 =± 62x = -12 ± 62

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