Problem on Factrorization and Finding the Factors for a Quadratic Equation

Topic : Factorization
Problem : Solve 8x^2 + 2x - 3

Solution :
8x^2 + 2x - 3
a = 8 ; b = 2 ; c = -3
ac = 8x * - 3 = -24x
Now we have find two numbers such that sum of those should be equal to +2x and product equal to -24x

6x -4x = +2x
So, 8x^2 + 2x - 3
8x^2 -4x+6x -3
4x(2x-1)+3(2x-1)
(2x-1)(4x+3) are the factors

A Simple Problem on Integration

Topic : Integration

Problem : Integrate x e^(-3x²)

Solution :

Solution of Trignometric Equation in Radians

Topic : Trignometric Equations

Question : Determine all solutions of each equation in radians in the interval (0,2 π)
4 Sin² x - 1 = 0

Solution :
4 Sin² x - 1 = 0
4 Sin² x = 1
Sin² x = 1/4
Sin x = √(1/4)
Sin x = ± 1/2

If Sin x = + 1/2
Sin x = Sin π/6
x = π/6
and
Sin x = Sin (π - π/6)
x = 5π/6

If Sin x = - 1/2
Sin x = Sin (π + π/6)
x = 7π/6
and
Sin x = Sin (2π - π/6)
x = 11π/6

Therefore the Answer is {π/6, 5π/6, 7π/6, 11π/6}

Question on Factorising the Function

Topic : Factorization

Questions : g(x) =X^3-4^2-3X+18 , factor g(x) into three linear factors. By setting each factor equal to zero, find the three x-intercepts. What do you notice about two intercepts?

Solution :
f (x) = x³ – 4x² - 3x + 18

we can see if x=3;

f(3)=3³ – 4(3)² – 3(3) +18
=27 -36 -9 + 18
= 45 – 45
=0
f(3) = 0 , x =3 is a zero of f(x)
(x-3) is a factor

Now divide f(x) by (x-3)

x-3)x³ – 4x² – 3x +18 (x² – x – 6
-----x³ – 3x²
--------------------------
------- - x² – 3x
------- - x² + 3x
--------------------------
----------- - 6x +18
----------- - 6x + 18
--------------------------
------------------ 0

x² – x – 6 is the quotient, factorize it,
x² – x – 6 = (x-3)(x+2)
Hence f(x) = (x-3)(x-3)(x+2) = (x-3)² (x+2)

Problem on Finding length of Rhombus Diagonal

Topic : Rhombus

Question : The area of a rhombus is 525 cm ^2 and one of the diagonals is 15 cm. Find
the other diagonal.

Solution :
length of one diagonal d1 = 15 cm
and length of the other diagonal be = d2
Area of the rhombus = ½ ( d1 x d2)
525 = ½ ( 15 x d2)
or 525 x 2 / 15 = d2
or d2 = 70 cm
Therefore, d2 = 70 cm

Question on Number of SailBoats

Topic : Sailboates Word Problem

Question : Use a ratio box to solve this problem.The ratio of sailboats to rowboats in the bay was 7 to 4.If there were 56 sailboats in the bay,how many rowboats were there'

Solution :

7:4 :: 56: x
7/4=56/x
X= 56 x4/7
X = 32
There fore there is 32 rowboats in a bay.