solve using quadratic formula
x^2-64x+576=0
Answers
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.
Solving a Single Variable Equation :
2.2 Solve : x+72 = 0
Subtract 72 from both sides of the equation :
x = -72
Solving a Single Variable Equation :
2.3 Solve : x-8 = 0
Add 8 to both sides of the equation :
x = 8
Supplement : Solving Quadratic Equation Directly
Solving x2+64x-576 = 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+64x-576
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 -32.0000
Plugging into the parabola formula -32.0000 for x we can calculate the y -coordinate :
y = 1.0 * -32.00 * -32.00 + 64.0 * -32.00 - 576.0
or y = -1600.000
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = x2+64x-576
Axis of Symmetry (dashed) {x}={-32.00}
Vertex at {x,y} = {-32.00,-1600.00}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-72.00, 0.00}
Root 2 at {x,y} = { 8.00, 0.00}
Solve Quadratic Equation by Completing The Square
3.2 Solving x2+64x-576 = 0 by Completing The Square .
Add 576 to both side of the equation :
x2+64x = 576
Now the clever bit: Take the coefficient of x , which is 64 , divide by two, giving 32 , and finally square it giving 1024
Add 1024 to both sides of the equation :
On the right hand side we have :
576 + 1024 or, (576/1)+(1024/1)
The common denominator of the two fractions is 1 Adding (576/1)+(1024/1) gives 1600/1
So adding to both sides we finally get :
x2+64x+1024 = 1600
Adding 1024 has completed the left hand side into a perfect square :
x2+64x+1024 =
(x+32) • (x+32) =
(x+32)2
Things which are equal to the same thing are also equal to one another. Since
x2+64x+1024 = 1600 and
x2+64x+1024 = (x+32)2
then, according to the law of transitivity,
(x+32)2 = 1600
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+32)2 is
(x+32)2/2 =
(x+32)1 =
x+32
Now, applying the Square Root Principle to Eq. #3.2.1 we get:
x+32 = √ 1600
Subtract 32 from both sides to obtain:
x = -32 + √ 1600
Since a square root has two values, one positive and the other negative
x2 + 64x - 576 = 0
has two solutions:
x = -32 + √ 1600
or
x = -32 - √ 1600
Solve Quadratic Equation using the Quadratic Formula
3.3 Solving x2+64x-576 = 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 = 64
C = -576
Accordingly, B2 - 4AC =
4096 - (-2304) =
6400
Applying the quadratic formula :
-64 ± √ 6400
x = ———————
2
Can √ 6400 be simplified ?
Yes! The prime factorization of 6400 is
2•2•2•2•2•2•2•2•5•5
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).
√ 6400 = √ 2•2•2•2•2•2•2•2•5•5 =2•2•2•2•5•√ 1 =
± 80 • √ 1 =
± 80
So now we are looking at:
x = ( -64 ± 80) / 2
Two real solutions:
x =(-64+√6400)/2=-32+40= 8.000 or:
x =(-64-√6400)/2=-32-40= -72.000
Two solutions were found :
x = 8
x = -72
so by quadratic formula
- b +- root b^2-4ac/2a
-(-64) +- root 64^2 - 4×1×( 576)/2×1
64 +- root 1792/2
64 +- 16 root 7 /2
answer will be 64+16 root 7/2 And 64 - 16 root 7 / 2
hope this answer will help u..