Factorise:- (2a+5a)(2a+5a-19)
Answers
Answer:
Step-by-step explanation:
The first term is, 2a2 its coefficient is 2 .
The middle term is, +5a its coefficient is 5 .
The last term, "the constant", is +19
Step-1 : Multiply the coefficient of the first term by the constant 2 • 19 = 38
Step-2 : Find two factors of 38 whose sum equals the coefficient of the middle term, which is 5 .
-38 + -1 = -39
-19 + -2 = -21
-2 + -19 = -21
-1 + -38 = -39
1 + 38 = 39
2 + 19 = 21
19 + 2 = 21
38 + 1 = 39
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step
3
:
-2a2 - 5a - 19 = 0
STEP
4
:
Parabola, Finding the Vertex:
4.1 Find the Vertex of y = -2a2-5a-19
Parabolas have a highest or a lowest point called the Vertex . Our parabola opens down and accordingly has a highest point (AKA absolute maximum) . We know this even before plotting "y" because the coefficient of the first term, -2 , is negative (smaller 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,Aa2+Ba+C,the a -coordinate of the vertex is given by -B/(2A) . In our case the a coordinate is -1.2500
Plugging into the parabola formula -1.2500 for a we can calculate the y -coordinate :
y = -2.0 * -1.25 * -1.25 - 5.0 * -1.25 - 19.0
or y = -15.875
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = -2a2-5a-19
Axis of Symmetry (dashed) {a}={-1.25}
Vertex at {a,y} = {-1.25,-15.88}
Function has no real roots
Solve Quadratic Equation by Completing The Square
4.2 Solving -2a2-5a-19 = 0 by Completing The Square .
Multiply both sides of the equation by (-1) to obtain positive coefficient for the first term:
2a2+5a+19 = 0 Divide both sides of the equation by 2 to have 1 as the coefficient of the first term :
a2+(5/2)a+(19/2) = 0
Subtract 19/2 from both side of the equation :
a2+(5/2)a = -19/2
Now the clever bit: Take the coefficient of a , which is 5/2 , divide by two, giving 5/4 , and finally square it giving 25/16
Add 25/16 to both sides of the equation :
On the right hand side we have :
-19/2 + 25/16 The common denominator of the two fractions is 16 Adding (-152/16)+(25/16) gives -127/16
So adding to both sides we finally get :
a2+(5/2)a+(25/16) = -127/16
Adding 25/16 has completed the left hand side into a perfect square :
a2+(5/2)a+(25/16) =
(a+(5/4)) • (a+(5/4)) =
(a+(5/4))2
Things which are equal to the same thing are also equal to one another. Since
a2+(5/2)a+(25/16) = -127/16 and
a2+(5/2)a+(25/16) = (a+(5/4))2
then, according to the law of transitivity,
(a+(5/4))2 = -127/16
We'll refer to this Equation as Eq. #4.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(a+(5/4))2 is
(a+(5/4))2/2 =
(a+(5/4))1 =
a+(5/4)
Now, applying the Square Root Principle to Eq. #4.2.1 we get:
a+(5/4) = √ -127/16
Subtract 5/4 from both sides to obtain:
a = -5/4 + √ -127/16
In Math, i is called the imaginary unit. It satisfies i2 =-1. Both i and -i are the square roots of -1
Since a square root has two values, one positive and the other negative
a2 + (5/2)a + (19/2) = 0
has two solutions:
a = -5/4 + √ 127/16 • i
or
a = -5/4 - √ 127/16 • i
Note that √ 127/16 can be written as
√ 127 / √ 16 which is √ 127 / 4
Solve Quadratic Equation using the Quadratic Formula
4.3 Solving -2a2-5a-19 = 0 by the Quadratic Formula .
According to the Quadratic Formula, a , the solution for Aa2+Ba+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
a = ————————
2A
In our case, A = -2
B = -5
C = -19
Accordingly, B2 - 4AC =
25 - 152 =
-127
Applying the quadratic formula :
5 ± √ -127
a = ——————
-4
In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written (a+b*i)
Both i and -i are the square roots of minus 1
Accordingly,√ -127 =
√ 127 • (-1) =
√ 127 • √ -1 =
± √ 127 • i
√ 127 , rounded to 4 decimal digits, is 11.2694
So now we are looking at:
a = ( 5 ± 11.269 i ) / -4
Two imaginary solutions :
a =(5+√-127)/-4=5/-4-i/4√ 127 = -1.2500+2.8174i
or:
a =(5-√-127)/-4=5/-4+i/4√ 127 = -1.2500-2.8174i