SOLVING QUADRATIC EQUATIONS NOT WRITTEN IN STANDARD FORM

Standard form of quadratic equation will be 

ax2 + bx + c = 0

where a, b and c are constants. Where a ≠ 0. 

Use the quadratic formula to solve exactly for x :

Problem 1 :

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

Solution :

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

x2  - x + 2x - 2 = 2 - 3x

x2 + x - 2 = 2 - 3x

Subtract 2 - 3x on each sides.

x2 + x - 2 - 2 + 3x = 0

x2 + 4x - 4 = 0

Using quadratic formula.

x = -b ± b2 - 4ac2ax = -4 ± (4)2 - 4(1)(-4)2(1)x = -4 ± 16 + 162(1)x = -4 ± 322x = -4 ± 422x = 2-2 ± 222x = -2 ± 22

Problem 2 :

(2x + 1)2 = 3 - x 

Solution :

(2x + 1)2 = 3 - x 

(2x)2 + 2(2x)(1) + 12 = 3 - x

4x2 + 4x + 1 = 3 - x

Subtract 3 - x on each sides.

4x2 + 4x + 1 - 3 + x  = 0

4x2 + 5x - 2 = 0

Using quadratic formula.

x = -b ± b2 - 4ac2ax = -5 ± (5)2 - 4(4)(-2)2(4)x = -5 ± 25 + 322(4)x = -5 ± 578x = -58 ± 578

Problem 3 :

(x - 2)2 = 1 + x 

Solution :

(x - 2)2 = 1 + x 

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

x2 - 4x + 4 = 1 + x

Subtract 1 + x on each sides.

x2 - 4x + 4 - 1 - x  = 0

x2 - 5x + 3 = 0

Using quadratic formula.

x = -b ± b2 - 4ac2ax = -(-5) ± (-5)2 - 4(1)(3)2(1)x = 5 ± 25 - 122(1)x = 5 ± 132x = 52 ± 132

Problem 4 :

x - 12 - x = 2x + 1

Solution :

x - 12 - x = 2x + 1

Using cross multiplication.

x - 1 = (2x + 1) (2 - x)

x - 1 = 4x - 2x2 + 2 - x

x - 1 = 3x - 2x2 + 2 

Subtract 3x - 2x2 + 2 on each sides.

x - 1 - 3x + 2x2 - 2 = 0

2x2 - 2x - 3 = 0

Using quadratic formula.

x = -b ± b2 - 4ac2ax = -(-2) ± (-2)2 - 4(2)(-3)2(2)x = 2 ± 4 + 242(2)x = 2 ± 284x = 2 ± 274x = 24 ± 274x = 12 ± 72

Problem 5 :

x - 1x = 1

Solution :

x - 1x = 1
x2 - 1x = 1

Using cross multiplication.

x2 - 1 = xx2 = x + 1x2 - x - 1 = 0

Using quadratic formula.

x = -b ± b2 - 4ac2ax = -(-1) ± (-1)2 - 4(1)(-1)2(1)x = 1 ± 1 + 42(1)x = 1 ± 52x = 12 ± 52

Problem 6 :

2x - 1x = 3

Solution :

2x - 1x = 3
2x2 - 1x = 3

Using cross multiplication.

2x2 - 1 = 3x2x2 - 1 - 3x = 0

Using quadratic formula.

x = -b ± b2 - 4ac2ax = -(-3) ± (-3)2 - 4(2)(-1)2(2)x = 3 ± 9 + 82(2)x = 3 ± 174x = 34 ± 174

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