Step by step solution :
Step 1 :
Trying to factor by splitting the middle term
1.1Factoring x2+24x-756
The first term is, x2 its coefficient is 1.
The middle term is, +24x its coefficient is 24.
The last term, "the constant", is -756
Step-1 : Multiply the coefficient of the first term by the constant 1•-756=-756
Step-2 : Find two factors of -756 whose sum equals the coefficient of the middle term, which is 24.
| -756 | + | 1 | = | -755 | ||
| -378 | + | 2 | = | -376 | ||
| -252 | + | 3 | = | -249 | ||
| -189 | + | 4 | = | -185 | ||
| -126 | + | 6 | = | -120 | ||
| -108 | + | 7 | = | -101 | ||
| -84 | + | 9 | = | -75 | ||
| -63 | + | 12 | = | -51 | ||
| -54 | + | 14 | = | -40 | ||
| -42 | + | 18 | = | -24 | ||
| -36 | + | 21 | = | -15 | ||
| -28 | + | 27 | = | -1 | ||
| -27 | + | 28 | = | 1 | ||
| -21 | + | 36 | = | 15 | ||
| -18 | + | 42 | = | 24 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -18 and 42
x2 - 18x+42x - 756
Step-4 : Add up the first 2 terms, pulling out like factors:
x•(x-18)
Add up the last 2 terms, pulling out common factors:
42•(x-18)
Step-5:Add up the four terms of step4:
(x+42)•(x-18)
Which is the desired factorization
Equation at the end of step 1 :
(x + 42) • (x - 18) = 0 Step 2 :
Theory - Roots of a product :
2.1 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:
2.2Solve:x+42 = 0Subtract 42 from both sides of the equation:
x = -42
Solving a Single Variable Equation:
2.3Solve:x-18 = 0Add 18 to both sides of the equation:
x = 18
Supplement : Solving Quadratic Equation Directly
Solving x2+24x-756 = 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:
3.1Find the Vertex ofy = x2+24x-756Parabolas 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,1, 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 -12.0000Plugging into the parabola formula -12.0000 for x we can calculate the y-coordinate:
y = 1.0 * -12.00 * -12.00 + 24.0 * -12.00 - 756.0
or y = -900.000
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = x2+24x-756
Axis of Symmetry (dashed) {x}={-12.00}
Vertex at {x,y} = {-12.00,-900.00}
x-Intercepts (Roots) :
Root 1 at {x,y} = {-42.00, 0.00}
Root 2 at {x,y} = {18.00, 0.00}
Solve Quadratic Equation by Completing The Square
3.2Solvingx2+24x-756 = 0 by Completing The Square.Add 756 to both side of the equation :
x2+24x = 756
Now the clever bit: Take the coefficient of x, which is 24, divide by two, giving 12, and finally square it giving 144
Add 144 to both sides of the equation :
On the right hand side we have:
756+144or, (756/1)+(144/1)
The common denominator of the two fractions is 1Adding (756/1)+(144/1) gives 900/1
So adding to both sides we finally get:
x2+24x+144 = 900
Adding 144 has completed the left hand side into a perfect square :
x2+24x+144=
(x+12)•(x+12)=
(x+12)2
Things which are equal to the same thing are also equal to one another. Since
x2+24x+144 = 900 and
x2+24x+144 = (x+12)2
then, according to the law of transitivity,
(x+12)2 = 900
We'll refer to this Equation as Eq. #3.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x+12)2 is
(x+12)2/2=
(x+12)1=
x+12
Now, applying the Square Root Principle to Eq.#3.2.1 we get:
x+12= √ 900
Subtract 12 from both sides to obtain:
x = -12 + √ 900
Since a square root has two values, one positive and the other negative
x2 + 24x - 756 = 0
has two solutions:
x = -12 + √ 900
or
x = -12 - √ 900
Solve Quadratic Equation using the Quadratic Formula
3.3Solvingx2+24x-756 = 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= 1
B= 24
C=-756 Accordingly,B2-4AC=
576 - (-3024) =
3600Applying the quadratic formula :
-24 ± √ 3600
x=———————
2Can √ 3600 be simplified ?
Yes!The prime factorization of 3600is
2•2•2•2•3•3•5•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).
√ 3600 =√2•2•2•2•3•3•5•5 =2•2•3•5•√ 1 =
±60 •√ 1 =
±60
So now we are looking at:
x=(-24±60)/2
Two real solutions:
x =(-24+√3600)/2=-12+30= 18.000
or:
x =(-24-√3600)/2=-12-30= -42.000
Two solutions were found :
- x = 18
- x = -42
