Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
0-(4*x^3+25*x^2-56*x)=0
Step by step solution :
Step 1 :
Equation at the end of step 1 :
0 - (((4 • (x3)) + 52x2) - 56x) = 0Step 2 :
Equation at the end of step 2 :
0 - ((22x3 + 52x2) - 56x) = 0 Step 3 :
Step 4 :
Pulling out like terms :
4.1 Pull out like factors:
-4x3 - 25x2 + 56x=-x•(4x2 + 25x - 56)
Trying to factor by splitting the middle term
4.2Factoring 4x2 + 25x - 56
The first term is, 4x2 its coefficient is 4.
The middle term is, +25x its coefficient is 25.
The last term, "the constant", is -56
Step-1 : Multiply the coefficient of the first term by the constant 4•-56=-224
Step-2 : Find two factors of -224 whose sum equals the coefficient of the middle term, which is 25.
| -224 | + | 1 | = | -223 | ||
| -112 | + | 2 | = | -110 | ||
| -56 | + | 4 | = | -52 | ||
| -32 | + | 7 | = | -25 | ||
| -28 | + | 8 | = | -20 | ||
| -16 | + | 14 | = | -2 | ||
| -14 | + | 16 | = | 2 | ||
| -8 | + | 28 | = | 20 | ||
| -7 | + | 32 | = | 25 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -7 and 32
4x2 - 7x+32x - 56
Step-4 : Add up the first 2 terms, pulling out like factors:
x•(4x-7)
Add up the last 2 terms, pulling out common factors:
8•(4x-7)
Step-5:Add up the four terms of step4:
(x+8)•(4x-7)
Which is the desired factorization
Equation at the end of step 4 :
-x • (4x - 7) • (x + 8) = 0 Step 5 :
Theory - Roots of a product :
5.1 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 separatelyIn other words, we are going to solve as many equations as there are terms in the productAny solution of term = 0 solves product = 0 as well.
Solving a Single Variable Equation:
5.2Solve:-x = 0Multiply both sides of the equation by (-1) : x = 0
Solving a Single Variable Equation:
5.3Solve:4x-7 = 0Add 7 to both sides of the equation:
4x = 7
Divide both sides of the equation by 4:
x = 7/4 = 1.750
Solving a Single Variable Equation:
5.4Solve:x+8 = 0Subtract 8 from both sides of the equation:
x = -8
Supplement : Solving Quadratic Equation Directly
Solving 4x2+25x-56 = 0 directly Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula
Parabola, Finding the Vertex:
6.1Find the Vertex ofy = 4x2+25x-56Parabolas 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 -3.1250Plugging into the parabola formula -3.1250 for x we can calculate the y-coordinate:
y = 4.0 * -3.12 * -3.12 + 25.0 * -3.12 - 56.0
or y = -95.062
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = 4x2+25x-56
Axis of Symmetry (dashed) {x}={-3.12}
Vertex at {x,y} = {-3.12,-95.06}
x-Intercepts (Roots) :
Root 1 at {x,y} = {-8.00, 0.00}
Root 2 at {x,y} = { 1.75, 0.00}
Solve Quadratic Equation by Completing The Square
6.2Solving4x2+25x-56 = 0 by Completing The Square.Divide both sides of the equation by 4 to have 1 as the coefficient of the first term :
x2+(25/4)x-14 = 0
Add 14 to both side of the equation :
x2+(25/4)x = 14
Now the clever bit: Take the coefficient of x, which is 25/4, divide by two, giving 25/8, and finally square it giving 625/64
Add 625/64 to both sides of the equation :
On the right hand side we have:
14+625/64or, (14/1)+(625/64)
The common denominator of the two fractions is 64Adding (896/64)+(625/64) gives 1521/64
So adding to both sides we finally get:
x2+(25/4)x+(625/64) = 1521/64
Adding 625/64 has completed the left hand side into a perfect square :
x2+(25/4)x+(625/64)=
(x+(25/8))•(x+(25/8))=
(x+(25/8))2
Things which are equal to the same thing are also equal to one another. Since
x2+(25/4)x+(625/64) = 1521/64 and
x2+(25/4)x+(625/64) = (x+(25/8))2
then, according to the law of transitivity,
(x+(25/8))2 = 1521/64
We'll refer to this Equation as Eq. #6.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x+(25/8))2 is
(x+(25/8))2/2=
(x+(25/8))1=
x+(25/8)
Now, applying the Square Root Principle to Eq.#6.2.1 we get:
x+(25/8)= √ 1521/64
Subtract 25/8 from both sides to obtain:
x = -25/8 + √ 1521/64
Since a square root has two values, one positive and the other negative
x2 + (25/4)x - 14 = 0
has two solutions:
x = -25/8 + √ 1521/64
or
x = -25/8 - √ 1521/64
Note that √ 1521/64 can be written as
√1521 / √64which is 39 / 8
Solve Quadratic Equation using the Quadratic Formula
6.3Solving4x2+25x-56 = 0 by the Quadratic Formula.According to the Quadratic Formula,x, the solution forAx2+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= 25
C=-56 Accordingly,B2-4AC=
625 - (-896) =
1521Applying the quadratic formula :
-25 ± √ 1521
x=———————
8Can √ 1521 be simplified ?
Yes!The prime factorization of 1521is
3•3•13•13
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).
√ 1521 =√3•3•13•13 =3•13•√ 1 =
±39 •√ 1 =
±39
So now we are looking at:
x=(-25±39)/8
Two real solutions:
x =(-25+√1521)/8=(-25+39)/8= 1.750
or:
x =(-25-√1521)/8=(-25-39)/8= -8.000
Three solutions were found :
- x = -8
- x = 7/4 = 1.750
- x = 0
