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Find the sum of the roots of the quadratic x^2+7x-13=0

Find all values of x such that 2x^2 +9x- 5 = 0

Find all solutions to the equation 10x^2 - 7x - 6 = 0

Find all values of x such that 6 = 35/x - 49/x^2

What is the sum of the roots of the quadratic 4x^2 - 4x - 4?

Find the sum of all solutions to the equation 3x(x+4)= 66

One root of x^2 + 12x + k = 0 is twice the other root. Find k.

Not Homework i swear im just showing model problems i recieved.

Dec 15, 2018

#1
+103915
+1

Find the sum of the roots of the quadratic x^2+7x-13=0

In the form ax^2 + bx + c, the sum of the roots =  -b / a

Here       a = 1     and b = 7

So....the sum of the roots  =   -7/1   =  -7

Find all values of x such that 2x^2 +9x- 5 = 0

We could use the quadratic formula to solve, but I like to see if this can factor, first

It will!!!

Factor as

(2x  - 1) ( x + 5)  = 0

Set each factor to 0  and solve for x

2x - 1  = 0                                                    x + 5 = 0

add 1 to both sides                                     subtract 5 from both sides

2x  = 1                                                           x = -5

divide both sides by 2

x = 1/2

The two solutions are in red

Dec 15, 2018
#2
+103915
+2

Find all solutions to the equation 10x^2 - 7x - 6 = 0

Again....let's see if we can factor

(5x  - 6) ( 2x + 1) = 0

Set both faxctors to x and solve for x

5x - 6 = 0                         2x + 1 = 0

5x = 6                              2x = - 1

x = 6/5                              x = -1/2

Find all values of x such that 6 = 35/x - 49/x^2

Here.....they are trying to be tricky.......multiply through by x^2   and we get

6x^2 =  35x - 49        rearrange as

6x^2 - 35x + 49= 0      see if we can factor

(3x - 7) (2x - 7)  = 0

Set both factors to 0   and solve for x

3x - 7  = 0                2x  - 7  = 0

3x  =  7                    2x   = 7

x =  7/3                      x = 7/2

Dec 15, 2018
#3
+103915
+2

What is the sum of the roots of the quadratic 4x^2 - 4x - 4?

Sum of the roots  =     - (-4) / 4   =   1

Find the sum of all solutions to the equation 3x(x+4)= 66

Simplify

3x^2 + 3x = 66

3x^2 + 3x - 66 = 0

Sum of the solutions (roots)   =   -3/3  = - 1

One root of x^2 + 12x + k = 0 is twice the other root. Find k.

This one is a little tricky.......

The sum of the roots  is    -12/ 1  =  -12

Let one root = a     and the other 2a

So.....the sum of the roots = -12

a + 2a  =  -12

3a = -12

a = -4

2a =  2(-4)  =  -8

Now.... in the form    ax^2 + bx + c

The product of the roots is     c /a

Here    a = 1    and     c = k

So.....the product of the roots   is       k /1    = k

So

(-4) (-8)   = k   =    32

Dec 15, 2018