Back to: Quadratic Equations
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Example #1
Using factorisation method , solve 2x² + 13x = 15
Solution
Steps to solve :
1. Move terms to the left side . 2x² + 13x - 15 = 0
2 . Multiply 2x² by -15 = -30x2
3. Find the factors of -30x²
- 1x and + 30x
lx and -30x
-2x and 15x
2x and -15x
-3x and 10x
3x and -10x
4 . Add the pairs of factors . The one that adds up to +13x among the six factors above is -2x and + 15x
-2 + 15x = +13x
5 . Replace the +13x in the equation given with -2x + 15x
2x² + 13x - 15 = 0
2x² -2x + 15x - 15 = 0
6. Factorize by taking the first 2 and the last 2
2x² -2x = 2x( x - 1 )
15x - 15 = +15 (x - 1)
7 . Combine the factorized terms .
2x ( x - 1 ) + 15 ( x - 1 ) = 0
Note : The values in the bracket must be the same . This is proof of being correct .
8 . Collect the terms outside the bracket
( 2x + 15 ) ( x - 1 ) = 0
9 . Equate each bracket to zero
2x + 15 = 0 and x - 1 = 0
10 Solve for x in each equation
2x + 15 = 0
2x = -15
x = -15 / 2 = -7.5
x = -7.5
x - 1 = 0
x = 1
Answer: x = 7.5 or 1Second Example #2
Factorize: 2a²+7ab-15b²
Steps to solve
1. Multiply 2a² and -15b² = - 30a²b².
2. Find the factors of -30a²b²
-1ab and 30ab = + 29ab ( sum )
1ab and -30ab -29ab ( Sum )
-2ab and 15ab = + 13ab ( sum )
2ab and -15ab = -13ab ( Sum )
-3ab and 10ab = +7ab ( Sum )
3ab and -10ab = -7ab ( sum )
-5ab and 6ab = +1ab ( sum )
5ab and -6ab = -1ab ( Sum )
3. Add the pairs of factors . The one that adds up to + 7ab among the eight pairs of factor above is - 3ab and + 10ab
-3ab + 10ab = + 7ab .
4. Replace the + 7ab in the equation given with -3ab and +10ab
2a² + 7ab - 15b²
2a² - 3ab +10ab - 15b²
5 Factorize by taking the first two terms and the last two terms
2a² - 3b = a(2a - 3b)
+10ab - 15b² = + 5b ( 2a - 3b) .
6 . Combine the factorized terms .
a (2a - 3b ) + 5b ( 2a - 3b )
Note : The values in the bracket must be the same .
7. Collect the terms outside the brackets
(a + 5b )( 2a - 3b )
Answer: (a + 5b )( 2a - 3b )Example #3
Factorise a²-17a+42 and solve for x
Solution
Steps to solve
1 . Multiply a² and +42 = + 42a²
2. Find the factors of + 42a²
1a and 42a = 43a ( sum )
-1a and -42a = -43a ( sum )
2a and 21 a = 23a ( Sum )
-2a and -21a = -23a ( Sum )
За and 14 a = 17a ( Sum )
-3a and -14a = -17a ( sum )
6a and 7а = 13a ( sum )
-6a and -7a = -13a ( sum )
3. Add the pairs of factors : The one that adds up to -17a among the eight factors above is -3a and -14a
- 3a + ( -14a ) = -17a .
4. Replace the -17a in the equation given with -3a - 14a
a²-17a +42 = 0
a² - 3a- 14a +42 = 0
5. Factorize by taking the first two parts and the last two parts.
a - 3a = a ( a - 3 )
- 14a +42 = -14 ( a - 3 )
6 . Combine the factorized terms .
a(a - 3 )-14 ( a - 3 ) = 0
Note : The values in the bracket must be the same
7. Collect the terms outside the brackets
( a - 14 ) ( a - 3 ) = 0
Note : Choose one of the two ( a - 3 ) since they are the same
8. Equate each bracket to zero
a-14 = 0 and a - 3 = 0
9. Solve for x in each equation.
a - 14 = 0
a = 14
a - 3 = 0
a = 3
Answer : x = 14 or 3Example #4
Using the factorization method , solve 7-22x + 3x²
Steps to solve
1. Multiply 7 and -3x² = + 21x²
Note : You may first rearrange into 3x² - 22 +7 or not .
2. Find the factors of + 21x²
+1x and 21x = + 22x ( sum )
- 1x and -21x = - 22x ( sum )
+ 3x and + 7x = + 10x ( sum )
-3x and -7x = -10x ( sum )
3. Add the pairs of factors . The one that adds up to -22x among the four pairs of factors above is -1x and -21x
- 1x + ( -21x ) = -22x
4. Replace the -22x in the equation given with -1x-21x
7- 22x + 3x²
7 - 1x - 21x + 3x²
5. Factorize by taking the first two terms and the last two terms .
7- 1x = 1 ( 7 - x )
- 2x + 3x² = -3x ( 7 - x )
6 Combine the factorized terms
1 ( 7-x ) -3x ( 7-x ) = 0
Note : The values in the bracket must be the same
7. Collect the terms outside the brackets
( 1-3x ) ( 7-x ) = 0
Note : Choose one of the two ( 7 - x ) since they are the same.
8. Equate each bracket to zero.
( 1-3x) = 0 and 7 - x = 0
9. Solve for x in each equation
1-3x = 0
1 = 3x
x = 1/3 and .
7-x = 0
7 = x
x = 7
Answer : x = 1/3 or 7Example #5
Solve the equation : a² - 3a = 0
Steps to solve
1. Factorize by collecting the letter common
a² - 3a = 0
a(a - 3 ) = 0
2. Equate the letter outside the bracket to zero and also the ones in the bracket .
a = O and a - 3 = 0
3. Solve for a
a = 0 and a = 3
Example #6
Example
Solve the equation m² = 16
Method 1
Steps to Solve
1. Take the root of both sides
√m² = √16
m = +/- 4
2. You may separate the answer
m = +4 or 4
Answer : M = +4 or -4
Method II
Steps to solve
1. Move the term '16' to the left .
m² - 16 = 0
Note : The above equation is lacking a term consisting of m . It must be that the coefficient is 0
i.e; m² + 0m -16 = 0
It can be re - written in this form .
2. Multiply m² and -16 = -16m²
3. Find the factors of -16m²
-1m and 16m = 15m ( sum )
1m and -16m = -15m ( sum )
-2m and 8m = 6m ( Sum )
2m and -8m = -6m ( Sum )
-4m and 4m = 0m ( Sum )
4. Add the pairs of factors . The one that adds 0m among the five pairs of factors above is -4m and 4m .
5. Replace the 0m in the equation given with -4m + 4m
m² + 0m - 16 = 0
m² - 4m + 4m - 16 = 0
6. Factorize by taking the first two terms and the last two terms
m² - 4m = m(m-4)
+4m - 16 = +4(m-4)
Note: The values in the brackets must be the same
7. Combine the factorized terms
m(m-4) +(m-4) = 0
8. Collect the terms outside the bracket and equate them to zero separately
(m+4)(m-4) = 0
m +4 = 0 and m-4=0
Answer: m= -4 or +4