
Question1
Solve the following quadratic equation.
x ^{2}  3x = 0
Solution to Question1

Given
x ^{2}  3x = 0

Factor x out in the expression on the left.
x (x  3) = 0

For the product x (x  3) to be equal to zero we nedd to have
x = 0 or x  3 = 0

Solve the above simple equations to obtain the solutions.
x = 0
or
x = 3

As an exercise, check that x = 0 and x = 3 are solutions to the given equation.
Question2
Solve the quadratic equation given below
x ^{2}  5 x + 6 = 0
Solution to Question2

To factor the expression on the left, we need to write x ^{2}  5 x + 6 in the form factored:
x ^{2}  5 x + 6 = (x + a)(x + b)

so that the sum of a and b is 5 and their product is 6. The numbers that satisfy these conditions are  2 and  3. Hence
x ^{2}  5 x + 6 = (x  2)(x  3)

Substitute into the original equation and solve.
(x  2)(x  3) = 0

(x  2)(x  3) is equal to zero if
x  2 = 0
or
x  3 = 0

Solve the above equations to obtain two solutions to the given equation.
x = 2
or
x = 3

As an exercise, check that x = 0 and x = 3 are solutions to the given equation.
Question3
Solve the following equation
2 x ^{2} + x  21 = 0
Solution to Question3

We first try to write 2 x ^{2} + x  21 in the factored form
2 x ^{2} + x  21 = (2x + a)(x + b)

Such that the product a b is equal to  21 and a + 2 b = 1
two pairs of numbers gives a product of  21: either 3 and 7 or 3 and 7. After some trial exercises it found that 2 x ^{2} + x  21 may be factored as follows:
2 x ^{2} + x  21 = (2x + 7)(x  3)

We now substitute into the original equation
(2x + 7)(x  3) = 0

and solve the following simpler equations
2x + 7 = 0
x  3 = 0

to obtain
x =  7 / 2
or x = 3

As an exercise, check that x = 0 and x = 3 are solutions to the given equation.
Question4
Solve the following equation
(x  1)(x + 1 / 2) =  x + 1
Solution to Question4

At first we might be tempted into expanding the left side of the equation. However after examination of the right side, the above equation may be written as:
(x  1)(x + 1 / 2) =  (x  1)

Write the equation with the right side equal to zero.
(x  1)(x + 1 / 2) + (x  1) = 0

We now factor (x  1) out.
(x  1)(x + 1 / 2 + 1) = 0

and solve the following simpler equations
x  1 = 0
x + 3 / 2 = 0

to obtain
x = 1
or
x =  3 / 2
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