Solve 8x^2-4x-15=0 by completing method
Answers
8x^2-4x-15=0
8x2-4x-15=0
Two solutions were found :
x =(4-√496)/16=(1-√ 31 )/4= -1.142
x =(4+√496)/16=(1+√ 31 )/4= 1.642
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(23x2 - 4x) - 15 = 0
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 8x2-4x-15
The first term is, 8x2 its coefficient is 8 .
The middle term is, -4x its coefficient is -4 .
The last term, "the constant", is -15
Step-1 : Multiply the coefficient of the first term by the constant 8 • -15 = -120
Step-2 : Find two factors of -120 whose sum equals the coefficient of the middle term, which is -4 .
-120 + 1 = -119
-60 + 2 = -58
-40 + 3 = -37
-30 + 4 = -26
-24 + 5 = -19
-20 + 6 = -14
-15 + 8 = -7
-12 + 10 = -2
-10 + 12 = 2
-8 + 15 = 7
-6 + 20 = 14
-5 + 24 = 19
-4 + 30 = 26
-3 + 40 = 37
-2 + 60 = 58
-1 + 120 = 119
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step 2 :
8x2 - 4x - 15 = 0
Step 3 :
Parabola, Finding the Vertex :
3.1 Find the Vertex of y = 8x2-4x-15
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, 8 , 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 0.2500
Plugging into the parabola formula 0.2500 for x we can calculate the y -coordinate :
y = 8.0 * 0.25 * 0.25 - 4.0 * 0.25 - 15.0
or y = -15.500
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = 8x2-4x-15
Axis of Symmetry (dashed) {x}={ 0.25}
Vertex at {x,y} = { 0.25,-15.50}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-1.14, 0.00}
Root 2 at {x,y} = { 1.64, 0.00}
Solve Quadratic Equation by Completing The Square
3.2 Solving 8x2-4x-15 = 0 by Completing The Square .
Divide both sides of the equation by 8 to have 1 as the coefficient of the first term :
x2-(1/2)x-(15/8) = 0
Add 15/8 to both side of the equation :
x2-(1/2)x = 15/8
Now the clever bit: Take the coefficient of x , which is 1/2 , divide by two, giving 1/4 , and finally square it giving 1/16
Add 1/16 to both sides of the equation :
On the right hand side we have :
15/8 + 1/16 The common denominator of the two fractions is 16 Adding (30/16)+(1/16) gives 31/16
So adding to both sides we finally get :
x2-(1/2)x+(1/16) = 31/16
Adding 1/16 has completed the left hand side into a perfect square :
x2-(1/2)x+(1/16) =
(x-(1/4)) • (x-(1/4)) =
(x-(1/4))2
Things which are equal to the same thing are also equal to one another. Since
x2-(1/2)x+(1/16) = 31/16 and
x2-(1/2)x+(1/16) = (x-(1/4))2
then, according to the law of transitivity,
(x-(1/4))2 = 31/16
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-(1/4))2 is
(x-(1/4))2/2 =
(x-(1/4))1 =
x-(1/4)
Now, applying the Square Root Principle to Eq. #3.2.1 we get:
x-(1/4) = √ 31/16
Add 1/4 to both sides to obtain:
x = 1/4 + √ 31/16
Since a square root has two values, one positive and the other negative
x2 - (1/2)x - (15/8) = 0
has two solutions:
x = 1/4 + √ 31/16
or
x = 1/4 - √ 31/16
Note that √ 31/16 can be written as
√ 31 / √ 16 which is √ 31 / 4
Solve Quadratic Equation using the Quadratic Formula
3.3 Solving 8x2-4x-15 = 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 = 8
B = -4
C = -15
Accordingly, B2 - 4AC =
16 - (-480) =
496
Applying the quadratic formula :
4 ± √ 496
x = —————
16
Can √ 496 be simplified ?
Yes! The prime factorization of 496 is
2•2•2•2•31
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).
√ 496 = √ 2•2•2•2•31 =2•2•√ 31 =
± 4 • √ 31
√ 31 , rounded to 4 decimal digits, is 5.5678
So now we are looking at:
x = ( 4 ± 4 • 5.568 ) / 16
Two real solutions:
x =(4+√496)/16=(1+√ 31 )/4= 1.642
or:
x =(4-√496)/16=(1-√ 31 )/4= -1.142
Two solutions were found :
x =(4-√496)/16=(1-√ 31 )/4= -1.142
x =(4+√496)/16=(1+√ 31 )/4= 1.642