4x²+6X+9=9 factoring pls
Answers
Solve Quadratic Equation using the Quadratic Formula
3.3 Solving 4x2-6x-9 = 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 = 4
B = -6
C = -9
Accordingly, B2 - 4AC =
36 - (-144) =
180
Applying the quadratic formula :
6 ± √ 180
x = —————
8
Can √ 180 be simplified ?
Yes! The prime factorization of 180 is
2•2•3•3•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).
√ 180 = √ 2•2•3•3•5 =2•3•√ 5 =
± 6 • √ 5
√ 5 , rounded to 4 decimal digits, is 2.2361
So now we are looking at:
x = ( 6 ± 6 • 2.236 ) / 8
Two real solutions:
x =(6+√180)/8=(3+3√ 5 )/4= 2.427
or:
x =(6-√180)/8=(3-3√ 5 )/4= -0.927
Two solutions were found :
x =(6-√180)/8=(3-3√ 5 )/4= -0.927
x =(6+√180)/8=(3+3√ 5 )/4= 2.427
Step by step solution :
STEP1:Equation at the end of step 1
(22x2 - 6x) - 9 = 0
STEP2:Trying to factor by splitting the middle term
2.1 Factoring 4x2-6x-9
The first term is, 4x2 its coefficient is 4 .
The middle term is, -6x its coefficient is -6 .
The last term, "the constant", is -9
Step-1 : Multiply the coefficient of the first term by the constant 4 • -9 = -36
Step-2 : Find two factors of -36 whose sum equals the coefficient of the middle term, which is -6 .
-36 + 1 = -35 -18 + 2 = -16 -12 + 3 = -9 -9 + 4 = -5 -6 + 6 = 0 -4 + 9 = 5 -3 + 12 = 9 -2 + 18 = 16 -1 + 36 = 35
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step2:
4x2 - 6x - 9 = 0
STEP3:Parabola, Finding the Vertex
3.1 Find the Vertex of y = 4x2-6x-9
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, 4 , 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 0.7500
Plugging into the parabola formula 0.7500 for x we can calculate the y -coordinate :
y = 4.0 * 0.75 * 0.75 - 6.0 * 0.75 - 9.0
or y = -11.250
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = 4x2-6x-9
Axis of Symmetry (dashed) {x}={ 0.75}
Vertex at {x,y} = { 0.75,-11.25}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-0.93, 0.00}
Root 2 at {x,y} = { 2.43, 0.00}