Solving Fractional Quadratic Equations
Cross-Multiplying =>
a = c => x bd a = c x bd => ad = bc
b d b d
[[ The Equation must be ax2+ bx + c = 0 before factorising ]]
Example
solve 6 = x + 3
x + 4
6 = x + 3
x + 4
6 = (x + 3)(x + 4) (Step1: Cross-Multiply)
6 = x2+ 7x + 12 (Step2: Expand to form a quadratic equation)
x2+ 7x + 12 – 6 = 0
x2+ 7x + 6 = 0
(x + 1)(x + 6) = 0 (Step3: Factorise and Solve for x)
x + 1 = 0 or x + 6 = 0
x = -1 or x = -6
Example
Solve 1 + 2 = 5
x -2 x – 3
(x – 3) + 2 x (x – 2) = 5 (Step1: LCM the denominator)
(x - 3)(x -2) (x – 2)(x – 3)
x – 3 + 2x – 4 = 5 (Step 2 : Combine to single fraction)
(x – 2)(x -3)
x + 2x – 3 – 4 = 5
(x – 2)(x -3)
3x – 7 = 5
(x – 2)(x – 3)
3x – 7 = 5(x – 2)(x – 3). (Step 3 : Cross-Multiply and Expand)
3x – 7 = 5(x2– 5x + 6)
3x – 7 = 5x2– 5x + 6
5x2– 5x + 6 – 3x + 7 = 0.
5x2-8x + 13 = 0 (Step 4: Factorise and Solve for x)
(5x + 13)(x + 1) = 0 5x \ / 13
5x + 13 = 0 or x + 1 = 0 x / \ 1
x = -13/5 or x = -1
Solving Equations by equating Coefficients
Example
Find the value of A, B and C
2x4 - 13x3 + 19x2 + 5x + 1 = (x - 4)(Ax + 1)(x2 + Bx + 1) + C
1. Such questions are usually solved by equating the coefficient values
2. If there are more than 2 brackets, expands in pairs
3. Solve the unit value first, and then the highest powers of X
Step 1 - Expand brackets (if needed)
2x4 - 13x3 + 19x2 + 5x + 1 = (x - 4)(Ax + 1)(x2 + Bx + 1) + C
2x4 - 13x3 + 19x2 + 5x + 1 = (x - 4)(Ax + 1)(x2 + Bx + 1) + C [Expands 1 pair]
(x - 4) (Ax + 1) = (Ax2 + (1 - 4)x - 4) (x2 + Bx + 1) + C
2x4 - 13x3 + 19x2 + 5x + 1 = (Ax2 + (1 - 4A)x - 4)(x2 + Bx + 1) + C
Step 2 - extract and solve from comparing coefficients
Extract coefficient of x4. [=> x2 x x2 , x3 x x , x4 x unit value ]
2x4 - 13x3 + 19x2 + 5x + 1 = (Ax2 + (1 - 4A)x - 4)(x2 + Bx + 1) + C
2x4 = Ax2 x x2
A = 2 [Equating x4 coefficient]
Extract and equating coefficient of unit value
1 = (-4 x 1) + C
C = 1 + 4
C = 5
Step 3 - Substitute values and find remaining unknown
2x4 - 13x3 + 19x2 + 5x + 1 = (2x2 -7x - 4)(x2 + Bx + 1) + 5
Extract and equating x coefficients
5x = -7x x 1 - 4Bx
5 = -7 - 4B
4B = -12
B = -3
A = 2, B = -3, C = 5
