What are the x-intercepts for the graph of y = x2 + 5x - 6?

We can use grouping to solve this quadratic because it can be factored.

x2+5x-6=0

x2 +6x-x-6=0

(x2+6x)-(x+6)=0

x(x+6)-(x+6)=0

(x-1)(x+6)=0

Next we can use the zero product property

x-1=0

x=1

or

x+6=0

x=-6

find a if a(a-8)=20

a(a-8)=20

so a= 10 or a=-2

Solve: (x-2) (x+4)=7

(x-2) (x+4)=7
x^2-2x+4x-8=7
x^2+2x-15=0
(x-3)(x+5)=0

Solution:
{3, -5}

If the square of a positive number is increased by seven times the number, the result is eight. Find the number.

1^2 + 7*1 =8 

...so 1 will work.

In general:
x^2+7x=8 leads to
x^2+7x-8=0
which factors to:
(x-1)(x+8)=0
but -8 is not positive

Solve: (x+9)^2 + 11(x+9)+18=0

2*9=18 and 2+9=11, therefore...
(x+9)^2 + 11(x+9)+18=
((x+9)+2)((x+9)+9)=
(x+11)(x+18)=0
Solution:
{-11, -18}

Solve this equation: d^2-5d+6=0

then...we'll use grouping to solve.

After looking at the factors 6 and 1 we notice that they don't quite work, so let's try 3 and 3:


Next we factor out the greatest common factors then we group them:


so that



Finally, because of the zero product property we can see that
x=3 or x=2 

 
We'll use grouping - there are many ways...but grouping works well for this problem 
 
 
 
 
or 
 
which leads to 
v=1/3 
or v=1

find the two positive integers required if the sum of a positive integer and its square is 30? find the number

so the number is 5

Solve the equation by using the square root property. Express the radical in simplest form: (m+12)^2=-9

 or 
 or 

a number added to eighteen times its reciprocal is eleven

x+18/x=11
x^2+18=11x
x^2-11x+18=0
(x-9)(x-2)=0 
x=2 or x=9

Write the quadratic equation in standard form

the sum of positive number and its square is 72.Find the number

x+x^2=72 
 
-8*9=-72 and -8+9=1 
so... 
 


checking...
-8+64 is not 72 
but 
9+63=72 
so it is 9

the hypotenuse of a right triangle is 24 ft long. the length of one leg is 6 feet more than the other. find the length of the legs.

c=24 
Clearly the side we know the least about is the shortest so... 
a=x 
the length of one leg is 6 feet more than the other means... 
b=x+6 
So our equation is: 
 
so here we go... 
 

 
divide by 2... 
 

but this quadratic cannot be factored - so we use the quadratic formula 
 


From which the negative solution can be discarded (why?) 
therefore... 
 
and so... 
{{a=3sqrt(31)-3}}} 
{{b=3sqrt(31)+3}}} 

Solve 4(x+2)^2+320=0

 

 

Square rooting both sides then subtracting 2 (from both sides) leads to 

 

or 

Solve for x (x - 2)(x + 1) = 4

Notice that we cannot apply the zero product property...yet!

(x - 2)(x + 1) = 4 
x^2-x-2-4=0 
x^2-x-6=0 
(x-3)(x+2)=0 
x=3 or x=-2

solve: 8x^2-3=2x

8x^2-3=2x 
8x^2-2x-3=0 
8x^2+4x-6x-3=0 
(8x^2+4x)-(6x+3)=0 
4x(2x+1)-3(2x+1)=0 
(4x-3)(2x+1)=0 

x=3/4 or x=-1/2

solve the following equation by graphing; x^2 + 4x = -3

 


which gives us x=-1 or x=-3

Solve and check 4n^2-1=0

 


 


 



 


 

so n=1/2 or n=-1/2 


u check, ok? 
:)

Use the square root property to solve the equation. If possible, simplify radicals or rationalize denominators. Express complex solutions in a+bi form

Equation:

Solution:

 

 
 
or 
 

one of my posts on Algebra.com

Solve: 231=7(x+8)x

231=7(x+8)x
the 231 seems large, so we look for a way to simplify - and it turns out to be a multiple of 7:



using cancellation and rearranging terms we get the following:




Notice that 11+(-3)=8 and (11)(-3)=-33 so we'll use them in our grouping:








which has solutions

x=3 and x=-11

:)