Example 1: 3x

Solution:

Example 2: 14x - 49x

Solution:

Example 3: Find the values of 'p' in the equation 4p

= 0

Solution:

Example 4: Find the solution for (a + b) for the equation: (a + b)

16(a + b) = -48

Solution:

Example 5: 12m

Solution:

^{2}+ 5x + 2 = 0Solution:

Since the given qaudratic equation is already in its standard form, we don't need

to convert it to the standard form. Factor the left hand side quadratic expression

by the method of splitting the middle term:

The master product is 3x

5x. Which two algebraic terms have the product of 6x

5x? They are 2x and 3x. Thus, the middle term 5x is split into two parts, 2x and

3x. Hence the equation becomes

factor out the common factors from each group. Hence the equation becomes

the middle term is done correctly on a non prime quadratic expression, there is

always a common factor between the two groups. Thus factoring out (x + 1), the equation

becomes

(x + 1)(3x + 2) = 0

The above obtained equation is now completely factored. Now we can apply the Zero

Product Rule to obtain the roots of the above equation. The Zero Product Rule states

that if the product of two quantities is zero, then either of the two quantities

is itslef equal to zero. Hence,

Either (x + 1) = 0, in which case x = -1

Or (3x + 2) = 0, in which case x = -2/3

Hence the roots of the given quadratic expression are -1 and -2/3.

to convert it to the standard form. Factor the left hand side quadratic expression

by the method of splitting the middle term:

The master product is 3x

^{2}* 2 = 6x^{2}and the middle term is5x. Which two algebraic terms have the product of 6x

^{2}and the sum of5x? They are 2x and 3x. Thus, the middle term 5x is split into two parts, 2x and

3x. Hence the equation becomes

3x

Now factor the left hand side by making two groups each containing two terms. Then^{2}+ 3x + 2x + 2 = 0factor out the common factors from each group. Hence the equation becomes

3x(x + 1) + 2(x + 1) = 0

(x + 1) is the common factor in the above expression. When the method of splittingthe middle term is done correctly on a non prime quadratic expression, there is

always a common factor between the two groups. Thus factoring out (x + 1), the equation

becomes

(x + 1)(3x + 2) = 0

The above obtained equation is now completely factored. Now we can apply the Zero

Product Rule to obtain the roots of the above equation. The Zero Product Rule states

that if the product of two quantities is zero, then either of the two quantities

is itslef equal to zero. Hence,

Either (x + 1) = 0, in which case x = -1

Or (3x + 2) = 0, in which case x = -2/3

Hence the roots of the given quadratic expression are -1 and -2/3.

Example 2: 14x - 49x

^{2}= -1Solution:

The given quadratic equation is not in its standard form, so converting it to standard

form:

49x

Now factor the left hand side expression by splitting its middle term. The master

product of the quadratic expression is -49x

Which two algebraic terms have the product of -49x

-7x and -7x are two terms whose product is -49x

the two parts of the middle term are -7x and -7x. Rewrite the quadratic equation

with the middle term split into the two parts:

49x

Factor the quadratic expression on the left hand side further by making two groups

- each group containing two terms - and factoring out the common factors from each

group. The resulting quadratic equation is

7x(7x - 1) - 1(7x - 1) = 0

(7x - 1) is the common factor between the two groups. Hence the result is,

(7x - 1)(7x - 1) = 0

That is,

(7x - 1)

Therefore 7x - 1 = 0, which implies that x = 1/7

Hence the given quadratic equation has two equal roots, 1/7 and 1/7.

form:

49x

^{2}- 14x + 1 = 0Now factor the left hand side expression by splitting its middle term. The master

product of the quadratic expression is -49x

^{2}and its middle term is -14x.Which two algebraic terms have the product of -49x

^{2}and the sum of -14x?-7x and -7x are two terms whose product is -49x

^{2}and sum is -14x. Thusthe two parts of the middle term are -7x and -7x. Rewrite the quadratic equation

with the middle term split into the two parts:

49x

^{2}- 7x - 7x + 1 = 0Factor the quadratic expression on the left hand side further by making two groups

- each group containing two terms - and factoring out the common factors from each

group. The resulting quadratic equation is

7x(7x - 1) - 1(7x - 1) = 0

(7x - 1) is the common factor between the two groups. Hence the result is,

(7x - 1)(7x - 1) = 0

That is,

(7x - 1)

^{2}= 0Therefore 7x - 1 = 0, which implies that x = 1/7

Hence the given quadratic equation has two equal roots, 1/7 and 1/7.

Example 3: Find the values of 'p' in the equation 4p

^{2}+ 12pq + 9q^{2}= 0

Solution:

The left hand side expression of this quadratic equation contains two variables.

The equation is such that it can be considered as written in either 'p' or 'q'.

But since the instructions clearly say that you have to find the values of 'p',

we will write the equation in 'p' only.

The given quadratic equation is already in the standard form with respect to the

variable 'p'. In order to factor its left hand side, split the middle term according

to the master product method.

The middle term of the quadratic expression on the left hand side is 12pq, and its

master product is 36p

product is equal to the master product and whose sum is equal to the middle term

are 6pq and 6pq. (6pq + 6pq = 12pq and 6pq * 6pq = 36p

Rewriting the equation with the middle term split into these two parts, we obtain

4p

Further factor the left hand side by grouping and factoring out common factors.

Thus the equation becomes,

2p(2p + 3q) + 3q(2p + 3q) = 0

Since (2p + 3q) is a common factor, factoring it out, the equation becomes

(2p + 3q)(2p + 3q) = 0

Thus 2p + 3q = 0, and hence p = -3q/2

Thus the given equation has equal two roots with respect to the variable 'p'. These

are -3q/2 and -3q/2.

The equation is such that it can be considered as written in either 'p' or 'q'.

But since the instructions clearly say that you have to find the values of 'p',

we will write the equation in 'p' only.

The given quadratic equation is already in the standard form with respect to the

variable 'p'. In order to factor its left hand side, split the middle term according

to the master product method.

The middle term of the quadratic expression on the left hand side is 12pq, and its

master product is 36p

^{2}q^{2}. Thus the two algebraic terms whoseproduct is equal to the master product and whose sum is equal to the middle term

are 6pq and 6pq. (6pq + 6pq = 12pq and 6pq * 6pq = 36p

^{2}q^{2}.Rewriting the equation with the middle term split into these two parts, we obtain

4p

^{2}+ 6pq + 6pq + 9q^{2}= 0Further factor the left hand side by grouping and factoring out common factors.

Thus the equation becomes,

2p(2p + 3q) + 3q(2p + 3q) = 0

Since (2p + 3q) is a common factor, factoring it out, the equation becomes

(2p + 3q)(2p + 3q) = 0

Thus 2p + 3q = 0, and hence p = -3q/2

Thus the given equation has equal two roots with respect to the variable 'p'. These

are -3q/2 and -3q/2.

Example 4: Find the solution for (a + b) for the equation: (a + b)

^{2}-16(a + b) = -48

Solution:

In this quadratic equation, the variable is not just a single variable, rather it

is an algebraic expression : (a + b). In order to view it as a single variable quadratic

equation, let y = (a + b), then the equation can be rewritten as

y

Rewrite the equation in the standard form (ax

y

Split the middle term of the quadratic expression on the left hand side. The middle

term is 16y and the master product is 48y

have the sum of 16y and the product of 48y

terms are 12y and 4y because 12y + 4y equals 16y and 12y times 4y equals 48y

Thus rewriting the equation with these two terms,

y

Factor the left hand side by grouping into two groups, one group containing the

first two terms and the other group containing the last two terms,

(y

Take common factors from each group:

y(y + 4) + 12(y + 4) = 0

Take (y + 4) as the common factor:

(y + 4)(y + 12) = 0

By applying the zero product rule, we obtain the roots of the quadratic equation,

Either y + 4 = 0 or y + 12 = 0

Either y = -4 or y = -12

Since y = a + b, therefore either (a + b) = -4 or (a + b) = -12

Therefore the roots of the given equation are -4 and -12.

is an algebraic expression : (a + b). In order to view it as a single variable quadratic

equation, let y = (a + b), then the equation can be rewritten as

y

^{2}+ 16y = -48Rewrite the equation in the standard form (ax

^{2}+ bx + c = 0),y

^{2}+ 16y + 48 = 0Split the middle term of the quadratic expression on the left hand side. The middle

term is 16y and the master product is 48y

^{2}. Which two algebraic termshave the sum of 16y and the product of 48y

^{2}? The two required algebraicterms are 12y and 4y because 12y + 4y equals 16y and 12y times 4y equals 48y

^{2}.Thus rewriting the equation with these two terms,

y

^{2}+ 4y + 12y + 48 = 0Factor the left hand side by grouping into two groups, one group containing the

first two terms and the other group containing the last two terms,

(y

^{2}+ 4y) + (12y + 48) = 0Take common factors from each group:

y(y + 4) + 12(y + 4) = 0

Take (y + 4) as the common factor:

(y + 4)(y + 12) = 0

By applying the zero product rule, we obtain the roots of the quadratic equation,

Either y + 4 = 0 or y + 12 = 0

Either y = -4 or y = -12

Since y = a + b, therefore either (a + b) = -4 or (a + b) = -12

Therefore the roots of the given equation are -4 and -12.

Example 5: 12m

^{2}+ m = 1Solution:

Rewrite the equation in the standard form:

12m

In order to factor the quadratic expression, split the middle term into two

parts such that their product is equal to the product of the first (ie 12m

of the quadratic expression.

12m

Make two groups of two terms each,

(12m

Factor out the common factors from each group,

4m(3m + 1) - 1(3m + 1) = 0

(3m + 1) is a common factor between the two groups, so factoring it out

(3m + 1)(4m - 1) = 0

By the Zero Product Rule, either (3m + 1) = 0 or (4m - 1) = 0

That is, either m = -1/3 or m = 1/4.

Hence the roots of the given equation are -1/3 and 1/4.

12m

^{2}+ m - 1 = 0In order to factor the quadratic expression, split the middle term into two

parts such that their product is equal to the product of the first (ie 12m

^{2}) and last (ie -1) termsof the quadratic expression.

12m

^{2}+ 4m - 3m - 1 = 0Make two groups of two terms each,

(12m

^{2}+ 4m) + (- 3m - 1) = 0Factor out the common factors from each group,

4m(3m + 1) - 1(3m + 1) = 0

(3m + 1) is a common factor between the two groups, so factoring it out

(3m + 1)(4m - 1) = 0

By the Zero Product Rule, either (3m + 1) = 0 or (4m - 1) = 0

That is, either m = -1/3 or m = 1/4.

Hence the roots of the given equation are -1/3 and 1/4.

superb :)

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