Please please please please please please help me with this queri i am confused
2 ✓2 x2 + 9x + 5 ✓2
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Step-by-step explanation:
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singhhjs13
09.06.2019
Math
Secondary School
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2 root 2x square - 9x - 5 root 2
2✓2x^2-9x-5✓2
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Answer:
Step-by-step explanation:
2x2-9x-5=0
Two solutions were found :
x = -1/2 = -0.500
x = 5
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(2x2 - 9x) - 5 = 0
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 2x2-9x-5
The first term is, 2x2 its coefficient is 2 .
The middle term is, -9x its coefficient is -9 .
The last term, "the constant", is -5
Step-1 : Multiply the coefficient of the first term by the constant 2 • -5 = -10
Step-2 : Find two factors of -10 whose sum equals the coefficient of the middle term, which is -9 .
-10 + 1 = -9 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -10 and 1
2x2 - 10x + 1x - 5
Step-4 : Add up the first 2 terms, pulling out like factors :
2x • (x-5)
Add up the last 2 terms, pulling out common factors :
1 • (x-5)
Step-5 : Add up the four terms of step 4 :
(2x+1) • (x-5)
Which is the desired factorization
Equation at the end of step 2 :
(x - 5) • (2x + 1) = 0
Step 3 :
Theory - Roots of a product :
3.1 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 :
3.2 Solve : x-5 = 0
Add 5 to both sides of the equation :
x = 5
Solving a Single Variable Equation :
3.3 Solve : 2x+1 = 0
Subtract 1 from both sides of the equation :
2x = -1
Divide both sides of the equation by 2:
x = -1/2 = -0.500
Supplement : Solving Quadratic Equation Directly
Solving 2x2-9x-5 = 0 directly
Parabola, Finding the Vertex :
Find the Vertex of y = 2x2-9x-5
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, 2 , 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 2.2500
Plugging into the parabola formula 2.2500 for x we can calculate the y -coordinate :
y = 2.0 * 2.25 * 2.25 - 9.0 * 2.25 - 5.0
or y = -15.125
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = 2x2-9x-5
Axis of Symmetry (dashed) {x}={ 2.25}
Vertex at {x,y} = { 2.25,-15.13}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-0.50, 0.00}
Root 2 at {x,y} = { 5.00, 0.00}
Solve Quadratic Equation by Completing The Square
4.2 Solving 2x2-9x-5 = 0 by Completing The Square .
Divide both sides of the equation by 2 to have 1 as the coefficient of the first term :
x2-(9/2)x-(5/2) = 0
Add 5/2 to both side of the equation :
x2-(9/2)x = 5/2
Now the clever bit: Take the coefficient of x , which is 9/2 , divide by two, giving 9/4 , and finally square it giving 81/16
Add 81/16 to both sides of the equation :
On the right hand side we have :
5/2 + 81/16 The common denominator of the two fractions is 16 Adding (40/16)+(81/16) gives 121/16
So adding to both sides we finally get :
x2-(9/2)x+(81/16) = 121/16
Adding 81/16 has completed the left hand side into a perfect square :
x2-(9/2)x+(81/16) =
(x-(9/4)) • (x-(9/4)) =
(x-(9/4))2
x2-(9/2)x+(81/16) = 121/16 and
x2-(9/2)x+(81/16) = (x-(9/4))2
then, according to the law of transitivity,
(x-(9/4))2 = 121/16
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
(x-(9/4))2 is
(x-(9/4))2/2 =
(x-(9/4))1 =
x-(9/4)
HOPE IT HELPS