Step by step solution :
Step 1 :
Equation at the kết thúc of step 1 :
(22x2 - 5x) - 12 = 0
Step 2 :
Trying vĩ đại factor by splitting the middle term
2.1 Factoring 4x2-5x-12
The first term is, 4x2 its coefficient is 4 .
The middle term is, -5x its coefficient is -5 .
The last term, "the constant", is -12
Bạn đang xem: 4x ^ 2 5x 12 = 0
Step-1 : Multiply the coefficient of the first term by the constant 4 • -12 = -48
Step-2 : Find two factors of -48 whose sum equals the coefficient of the middle term, which is -5 .
| -48 | + | 1 | = | -47 | ||
| -24 | + | 2 | = | -22 | ||
| -16 | + | 3 | = | -13 | ||
| -12 | + | 4 | = | -8 | ||
| -8 | + | 6 | = | -2 | ||
| -6 | + | 8 | = | 2 | ||
| -4 | + | 12 | = | 8 | ||
| -3 | + | 16 | = | 13 | ||
| -2 | + | 24 | = | 22 | ||
| -1 | + | 48 | = | 47 |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the kết thúc of step 2 :
4x2 - 5x - 12 = 0
Step 3 :
Parabola, Finding the Vertex :
3.1 Find the Vertex of y = 4x2-5x-12Parabolas 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 vãn 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 vĩ đại be able vĩ đại 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 0.6250 Plugging into the parabola formula 0.6250 for x we can calculate the y -coordinate :
y = 4.0 * 0.62 * 0.62 - 5.0 * 0.62 - 12.0
or y = -13.562
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = 4x2-5x-12
Axis of Symmetry (dashed) {x}={ 0.62}
Vertex at {x,y} = { 0.62,-13.56}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-1.22, 0.00}
Root 2 at {x,y} = { 2.47, 0.00}
Solve Quadratic Equation by Completing The Square
3.2 Solving 4x2-5x-12 = 0 by Completing The Square .Divide both sides of the equation by 4 to have 1 as the coefficient of the first term :
x2-(5/4)x-3 = 0
Add 3 vĩ đại both side of the equation :
x2-(5/4)x = 3
Now the clever bit: Take the coefficient of x , which is 5/4 , divide by two, giving 5/8 , and finally square it giving 25/64
Add 25/64 vĩ đại both sides of the equation :
On the right hand side we have :
3 + 25/64 or, (3/1)+(25/64)
The common denominator of the two fractions is 64 Adding (192/64)+(25/64) gives 217/64
So adding vĩ đại both sides we finally get :
x2-(5/4)x+(25/64) = 217/64
Adding 25/64 has completed the left hand side into a perfect square :
x2-(5/4)x+(25/64) =
(x-(5/8)) • (x-(5/8)) =
(x-(5/8))2
Things which are equal vĩ đại the same thing are also equal vĩ đại one another. Since
x2-(5/4)x+(25/64) = 217/64 and
x2-(5/4)x+(25/64) = (x-(5/8))2
then, according vĩ đại the law of transitivity,
(x-(5/8))2 = 217/64
We'll refer vĩ đại this Equation as Eq. #3.2.1
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The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x-(5/8))2 is
(x-(5/8))2/2 =
(x-(5/8))1 =
x-(5/8)
Now, applying the Square Root Principle to Eq. #3.2.1 we get:
x-(5/8) = √ 217/64
Add 5/8 vĩ đại both sides vĩ đại obtain:
x = 5/8 + √ 217/64
Since a square root has two values, one positive and the other negative
x2 - (5/4)x - 3 = 0
has two solutions:
x = 5/8 + √ 217/64
or
x = 5/8 - √ 217/64
Note that √ 217/64 can be written as
√ 217 / √ 64 which is √ 217 / 8
Solve Quadratic Equation using the Quadratic Formula
3.3 Solving 4x2-5x-12 = 0 by the Quadratic Formula .According vĩ đại 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 = -5
C = -12 Accordingly, B2 - 4AC =
25 - (-192) =
217Applying the quadratic formula :
5 ± √ 217
x = —————
8 √ 217 , rounded vĩ đại 4 decimal digits, is 14.7309
So now we are looking at:
x = ( 5 ± 14.731 ) / 8
Two real solutions:
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x =(5+√217)/8= 2.466
or:
x =(5-√217)/8=-1.216
Two solutions were found :
- x =(5-√217)/8=-1.216
- x =(5+√217)/8= 2.466
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