SOLUTION: 1. Rewrite y = x2 + 14x + 29 in general form. 2. Rewrite y = 3x2 - 24x + 10 in general form. 3. Solve for x: (x - 9)2 = 1 4. Solve for x: x2 + 24x + 90 = 0 5. Solve for x:

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: 1. Rewrite y = x2 + 14x + 29 in general form. 2. Rewrite y = 3x2 - 24x + 10 in general form. 3. Solve for x: (x - 9)2 = 1 4. Solve for x: x2 + 24x + 90 = 0 5. Solve for x:       Log On


   

Question 462262: 1. Rewrite y = x2 + 14x + 29 in general form.
2. Rewrite y = 3x2 - 24x + 10 in general form.
3. Solve for x: (x - 9)2 = 1
4. Solve for x: x2 + 24x + 90 = 0
5. Solve for x: 2x2 - 4x - 14 = 0
6. Create your own quadratic equation and demonstrate how it would be solved by graphing, factoring, the quadratic formula, and by completing the square.

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
first: Quadratic Functions in General Form is f%28x%29+=+ax2+%2B+bx+%2B+c or y+=+ax2+%2B+bx+%2B+c


1.y+=+x%5E2+%2B+14x+%2B+29...already in general form.

2. y+=+3x%5E2+-+24x+%2B+10...already in general form.
3. Solve for x:
%28x+-+9%29%5E2+=+1
%28sqrt%28x+-+9%29%5E2%29+=+sqrt%281%29
x+-+9+=+1
x+=+1%2B9

x+=+10


4. Solve for x:
x%5E2+%2B+24x+%2B+90+=+0+....use quadratic formula
Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve x%5E2%2B24%2Ax%2B90=0 ( notice a=1, b=24, and c=90)





x+=+%28-24+%2B-+sqrt%28+%2824%29%5E2-4%2A1%2A90+%29%29%2F%282%2A1%29 Plug in a=1, b=24, and c=90




x+=+%28-24+%2B-+sqrt%28+576-4%2A1%2A90+%29%29%2F%282%2A1%29 Square 24 to get 576




x+=+%28-24+%2B-+sqrt%28+576%2B-360+%29%29%2F%282%2A1%29 Multiply -4%2A90%2A1 to get -360




x+=+%28-24+%2B-+sqrt%28+216+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)




x+=+%28-24+%2B-+6%2Asqrt%286%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28-24+%2B-+6%2Asqrt%286%29%29%2F2 Multiply 2 and 1 to get 2


So now the expression breaks down into two parts


x+=+%28-24+%2B+6%2Asqrt%286%29%29%2F2 or x+=+%28-24+-+6%2Asqrt%286%29%29%2F2



Now break up the fraction



x=-24%2F2%2B6%2Asqrt%286%29%2F2 or x=-24%2F2-6%2Asqrt%286%29%2F2



Simplify



x=-12%2B3%2Asqrt%286%29 or x=-12-3%2Asqrt%286%29



So the solutions are:

x=-12%2B3%2Asqrt%286%29 or x=-12-3%2Asqrt%286%29





5. Solve for x:
2x%5E2+-+4x+-+14+=+0....use quadratic formula
Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve 2%2Ax%5E2-4%2Ax-14=0 ( notice a=2, b=-4, and c=-14)





x+=+%28--4+%2B-+sqrt%28+%28-4%29%5E2-4%2A2%2A-14+%29%29%2F%282%2A2%29 Plug in a=2, b=-4, and c=-14




x+=+%284+%2B-+sqrt%28+%28-4%29%5E2-4%2A2%2A-14+%29%29%2F%282%2A2%29 Negate -4 to get 4




x+=+%284+%2B-+sqrt%28+16-4%2A2%2A-14+%29%29%2F%282%2A2%29 Square -4 to get 16 (note: remember when you square -4, you must square the negative as well. This is because %28-4%29%5E2=-4%2A-4=16.)




x+=+%284+%2B-+sqrt%28+16%2B112+%29%29%2F%282%2A2%29 Multiply -4%2A-14%2A2 to get 112




x+=+%284+%2B-+sqrt%28+128+%29%29%2F%282%2A2%29 Combine like terms in the radicand (everything under the square root)




x+=+%284+%2B-+8%2Asqrt%282%29%29%2F%282%2A2%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%284+%2B-+8%2Asqrt%282%29%29%2F4 Multiply 2 and 2 to get 4


So now the expression breaks down into two parts


x+=+%284+%2B+8%2Asqrt%282%29%29%2F4 or x+=+%284+-+8%2Asqrt%282%29%29%2F4



Now break up the fraction



x=%2B4%2F4%2B8%2Asqrt%282%29%2F4 or x=%2B4%2F4-8%2Asqrt%282%29%2F4



Simplify



x=1%2B2%2Asqrt%282%29 or x=1-2%2Asqrt%282%29



So the solutions are:

x=1%2B2%2Asqrt%282%29 or x=1-2%2Asqrt%282%29




6. Create your own quadratic equation and demonstrate how it would be solved by graphing, factoring, the quadratic formula, and by completing the square.

y=x%5E2+%2B+3x+%2B+2....use quadratic formula
Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve x%5E2%2B3%2Ax%2B2=0 ( notice a=1, b=3, and c=2)





x+=+%28-3+%2B-+sqrt%28+%283%29%5E2-4%2A1%2A2+%29%29%2F%282%2A1%29 Plug in a=1, b=3, and c=2




x+=+%28-3+%2B-+sqrt%28+9-4%2A1%2A2+%29%29%2F%282%2A1%29 Square 3 to get 9




x+=+%28-3+%2B-+sqrt%28+9%2B-8+%29%29%2F%282%2A1%29 Multiply -4%2A2%2A1 to get -8




x+=+%28-3+%2B-+sqrt%28+1+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)




x+=+%28-3+%2B-+1%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28-3+%2B-+1%29%2F2 Multiply 2 and 1 to get 2


So now the expression breaks down into two parts


x+=+%28-3+%2B+1%29%2F2 or x+=+%28-3+-+1%29%2F2


Lets look at the first part:


x=%28-3+%2B+1%29%2F2


x=-2%2F2 Add the terms in the numerator

x=-1 Divide


So one answer is

x=-1




Now lets look at the second part:


x=%28-3+-+1%29%2F2


x=-4%2F2 Subtract the terms in the numerator

x=-2 Divide


So another answer is

x=-2


So our solutions are:

x=-1 or x=-2




y=x%5E2+%2B+3x+%2B+2....use graphing
find first several points that lie on a parabola
if x=0...->...y=2

if x=1...->...y=6
if x=-1...->...y=0
if x=-2...->...y=0
if x=-3...->...y=2
+graph%28+500%2C+500%2C+-10%2C+10%2C+-10%2C+10%2C+x%5E2+%2B+3x+%2B+2%29+

y=x%5E2+%2B+3x+%2B+2....use factoring
Solved by pluggable solver: Factoring Quadratics with a leading coefficient of 1 (a=1)
In order to factor 1%2Ax%5E2%2B3%2Ax%2B2, first we need to ask ourselves: What two numbers multiply to 2 and add to 3? Lets find out by listing all of the possible factors of 2


Factors:

1,2,

-1,-2,List the negative factors as well. This will allow us to find all possible combinations

These factors pair up to multiply to 2.

1*2=2

(-1)*(-2)=2

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to 3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 3

||
First Number|Second Number|Sum
1|2|1+2=3
-1|-2|-1+(-2)=-3
We can see from the table that 1 and 2 add to 3. So the two numbers that multiply to 2 and add to 3 are: 1 and 2 Now we substitute these numbers into a and b of the general equation of a product of linear factors which is: %28x%2Ba%29%28x%2Bb%29substitute a=1 and b=2 So the equation becomes: (x+1)(x+2) Notice that if we foil (x+1)(x+2) we get the quadratic 1%2Ax%5E2%2B3%2Ax%2B2 again


y=x%5E2+%2B+3x+%2B+2....use completing the square
convert it into y=%281%2F5%29x%5E2%2B%281%2F2%29x%2B1%2F10