# 49v^2-75v=-24-5vfactoring

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49v^2 - 75v = -24 - 5v

First we will group similar terms by moving all terms to the left side:

==> 49v^2 - 75 v + 5v + 24 = 0

==> 49v^2 - 70 v +24 = 0

==> ( 7v - 4) ( 7v - 6) = 0

==> v1 = 4/7

==> v2= 6/7

To factor 49v^2-75v=-24-5v, follow these steps:

49v^2-75v=-24-5v

=> 49v^2-75v+5v+24=0

=> 49v^2-70v+24=0

=>49v^2-42v-28v+24=0

=>7v(7v-6)-4(7v-6)=0

=>(7v-4)*(7v-6)=0

This can also be used to determine that v=4/7 and v=6/7

**We'll use another method to solve the quadratic equation.**

First, we'll have to re-write the equation.

We'll add 5v both sides:

49v^2 - 75v + 5v = -24

Now, we'll add 24 both sides:

49v^2 - 75v + 5v + 24 = 0

We'll combine like terms:

49v^2 - 70v + 24 = 0

From this point, we can calculate the roots using 2 methods

**First method**:

We'll apply the quadratic formula:

v1 = [-b+sqrt(b^2 - 4ac)]/2a

v1 = [70+sqrt(196)]/98

v1 = (70+14)/98

v1 = 84/98

v1 = 42/49

**v1 = 6/7**

v2 = (70-14)/98

v2 = 56/98

v2 = 28/49

**v2 = 4/7**

**Second method:**

We'll complete the square

49v^2 - 70v + 24 = 0

[(7v)^2 - 7*2*5v + 5^2] - 5^2 + 24 = 0

(7v-5)^2 - 1 = 0

We'll solve the difference of squares using the formula:

a^2 - b^2 = (a-b)(a+b)

(7v-5)^2 - 1 = (7v-5-1)(7v-5+1)

(7v-5)^2 - 1 = (7v-6)(7v-4)

But, (7v-5)^2 - 1 = 0, so (7v-6)(7v-4) = 0

We'll set each factor as 0:

7v - 6 = 0

We'll add 6 both sides:

7v = 6

We'll divide by 7:

**v = 6/7**

7v-4 = 0

We'll add 4 both sides:

7v = 4

**v = 4/7**

49v^2 - 75v = -24-5v. This is only an equation. We factor an expression, not an equation.

This is a quadratic equation. We can write it as a standard quadratic equation by simple operations of adding or subracting equal to both sides and rewrite as:

49v^ - 75v +24+5v = -24-5v+24+5v

49v^2 - 70v +24 = 0.

Regroup by splitting -70v = -42v abd -28v in order to factor the left side.

49v^2-42V - 28v +24 = 0

7v(7v-6) - 4(7v -6) = 0

(7v-6)(7v-4) = 0.

So by zero product rule 7v-6 = 0 or 7v-4 = 0

7v-6 = 0 gves: v = 6/7.

7v-4 = 0 gives: v = 4/7.

Therefore v = 6/7 or 4/7 are the solutions.