Question

Can someone please tell me the vertex and x intercepts of this equation;

Y = -16x^2 + 190x + 690,
Because I am coming with outrageous numbers and I am growing very impatient, I would just like to see what you guys are getting.

Answer 1

heres sa simlilar one 2x2-190x+1232=0

Two solutions were found :

x = 88
x = 7
Step by step solution :

Step 1 :

Simplify 2x2-190x + 1232
Pulling out like terms :

1.1 Pull out like factors :

2x2 - 190x + 1232 = 2 • (x2 - 95x + 616)

Trying to factor by splitting the middle term

1.2 Factoring x2 - 95x + 616

The first term is, x2 its coefficient is 1 .
The middle term is, -95x its coefficient is -95 .
The last term, "the constant", is +616

Step-1 : Multiply the coefficient of the first term by the constant 1 • 616 = 616

Step-2 : Find two factors of 616 whose sum equals the coefficient of the middle term, which is -95 .

-616 + -1 = -617
-308 + -2 = -310
-154 + -4 = -158
-88 + -7 = -95 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -88 and -7
x2 - 88x - 7x - 616

Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-88)
Add up the last 2 terms, pulling out common factors :
7 • (x-88)
Step-5 : Add up the four terms of step 4 :
(x-7) • (x-88)
Which is the desired factorization

Equation at the end of step 1 :

2 • (x - 7) • (x - 88) = 0
Step 2 :

Solve 2•(x-7)•(x-88) = 0
Theory - Roots of a product :

2.1 A product of several terms equals zero.

When a product of two or more terms equals zero, then at least one of the terms must be zero.

We shall now solve each term = 0 separately

In other words, we are going to solve as many equations as there are terms in the product

Any solution of term = 0 solves product = 0 as well.

Equations which are never true :

2.2 Solve : 2 = 0

This equation has no solution.
A a non-zero constant never equals zero.

Solving a Single Variable Equation :

2.3 Solve : x-7 = 0

Add 7 to both sides of the equation :
x = 7

Solving a Single Variable Equation :

2.4 Solve : x-88 = 0

Add 88 to both sides of the equation :
x = 88

Supplement : Solving Quadratic Equation Directly

Solving x2-95x+616 = 0 directly
Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula

Parabola, Finding the Vertex :

3.1 Find the Vertex of y = x2-95x+616

Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 1 , is positive (greater than zero).

Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.

Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.

For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is 47.5000

Plugging into the parabola formula 47.5000 for x we can calculate the y -coordinate :
y = 1.0 * 47.50 * 47.50 - 95.0 * 47.50 + 616.0
or y = -1640.250

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = x2-95x+616
Axis of Symmetry (dashed) {x}={47.50}
Vertex at {x,y} = {47.50,-1640.25}
x -Intercepts (Roots) :
Root 1 at {x,y} = { 7.00, 0.00}
Root 2 at {x,y} = {88.00, 0.00}



Solve Quadratic Equation by Completing The Square

3.2 Solving x2-95x+616 = 0 by Completing The Square .

Subtract 616 from both side of the equation :
x2-95x = -616

Now the clever bit: Take the coefficient of x , which is 95 , divide by two, giving 95/2 , and finally square it giving 9025/4

Add 9025/4 to both sides of the equation :
On the right hand side we have :
-616 + 9025/4 or, (-616/1)+(9025/4)
The common denominator of the two fractions is 4 Adding (-2464/4)+(9025/4) gives 6561/4
So adding to both sides we finally get :
x2-95x+(9025/4) = 6561/4

Adding 9025/4 has completed the left hand side into a perfect square :
x2-95x+(9025/4) =
(x-(95/2)) • (x-(95/2)) =
(x-(95/2))2
Things which are equal to the same thing are also equal to one another. Since
x2-95x+(9025/4) = 6561/4 and
x2-95x+(9025/4) = (x-(95/2))2
then, according to the law of transitivity,
(x-(95/2))2 = 6561/4

We'll refer to this Equation as Eq. #3.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
(x-(95/2))2 is
(x-(95/2))2/2 =
(x-(95/2))1 =
x-(95/2)

Now, applying the Square Root Principle to Eq. #3.2.1 we get:
x-(95/2) = √ 6561/4

Add 95/2 to both sides to obtain:
x = 95/2 + √ 6561/4

Since a square root has two values, one positive and the other negative
x2 - 95x + 616 = 0
has two solutions:
x = 95/2 + √ 6561/4
or
x = 95/2 - √ 6561/4

Note that √ 6561/4 can be written as
√ 6561 / √ 4 which is 81 / 2

Solve Quadratic Equation using the Quadratic Formula

3.3 Solving x2-95x+616 = 0 by the Quadratic Formula .

According to the Quadratic Formula, x , the solution for Ax2+Bx+C = 0 , where A, B and C are numbers, often called coefficients, is given by :

- B ± √ B2-4AC
x = ————————
2A

In our case, A = 1
B = -95
C = 616

Accordingly, B2 - 4AC =
9025 - 2464 =
6561

Applying the quadratic formula :

95 ± √ 6561
x = ——————
2

Can √ 6561 be simplified ?

Yes! The prime factorization of 6561 is
3•3•3•3•3•3•3•3
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).

√ 6561 = √ 3•3•3•3•3•3•3•3 =3•3•3•3•√ 1 =
± 81 • √ 1 =
± 81

So now we are looking at:
x = ( 95 ± 81) / 2

Two real solutions:

x =(95+√6561)/2=(95+81)/2= 88.000

or:

x =(95-√6561)/2=(95-81)/2= 7.000


Two solutions were found :

x = 88
x = 7

Processing ends successfully

Answer 2

Those question marks were throwing me off and I could not focus what so ever on what you were saying, I'm sorry.

Answer 3

\[y=-16 x^2+190x+690\]
\[y=-16\left( x^2-\frac{ 190 }{ 16 }x \right)+690=-16\left\{ x^2-\frac{ 190 }{ 16 }x+\left( \frac{ 190 }{ 2*16 } \right)^2-\left( \frac{ 190 }{ 2*16 } \right)^2\ \right\}+690\]
\[y=-16\left( x-\frac{ 190 }{ 2*16 } \right)^2+\frac{ 36100 }{ 1024 }+690\]
\[\left( x-\frac{ 95 }{ 16 } \right)^2=-\frac{ 1 }{ 16 }\left( y-\frac{ 36100+690*1024 }{ 1024 } \right)\]
it is a downward parabola.
vertex is \[\left( \frac{ 95 }{ 16 },\frac{ 36100+690*1024 }{ 1024 } \right)\]
you can simplify it.