Solution - Quadratic equations
Other Ways to Solve:
Step by Step Solution
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((32•7x2) + 59x) - 22 = 0
Step 2 :
Trying to factor by splitting the middle term
2.1 Factoring 63x2+59x-22
The first term is, 63x2 its coefficient is 63 .
The middle term is, +59x its coefficient is 59 .
The last term, "the constant", is -22
Step-1 : Multiply the coefficient of the first term by the constant 63 • -22 = -1386
Step-2 : Find two factors of -1386 whose sum equals the coefficient of the middle term, which is 59 .
-1386 | + | 1 | = | -1385 | ||
-693 | + | 2 | = | -691 | ||
-462 | + | 3 | = | -459 | ||
-231 | + | 6 | = | -225 | ||
-198 | + | 7 | = | -191 | ||
-154 | + | 9 | = | -145 | ||
-126 | + | 11 | = | -115 | ||
-99 | + | 14 | = | -85 | ||
-77 | + | 18 | = | -59 | ||
-66 | + | 21 | = | -45 | ||
-63 | + | 22 | = | -41 | ||
-42 | + | 33 | = | -9 | ||
-33 | + | 42 | = | 9 | ||
-22 | + | 63 | = | 41 | ||
-21 | + | 66 | = | 45 | ||
-18 | + | 77 | = | 59 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -18 and 77
63x2 - 18x + 77x - 22
Step-4 : Add up the first 2 terms, pulling out like factors :
9x • (7x-2)
Add up the last 2 terms, pulling out common factors :
11 • (7x-2)
Step-5 : Add up the four terms of step 4 :
(9x+11) • (7x-2)
Which is the desired factorization
Equation at the end of step 2 :
(7x - 2) • (9x + 11) = 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 : 7x-2 = 0
Add 2 to both sides of the equation :
7x = 2
Divide both sides of the equation by 7:
x = 2/7 = 0.286
Solving a Single Variable Equation :
3.3 Solve : 9x+11 = 0
Subtract 11 from both sides of the equation :
9x = -11
Divide both sides of the equation by 9:
x = -11/9 = -1.222
Supplement : Solving Quadratic Equation Directly
Solving 63x2+59x-22 = 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 :
4.1 Find the Vertex of y = 63x2+59x-22
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, 63 , 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 -0.4683
Plugging into the parabola formula -0.4683 for x we can calculate the y -coordinate :
y = 63.0 * -0.47 * -0.47 + 59.0 * -0.47 - 22.0
or y = -35.813
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = 63x2+59x-22
Axis of Symmetry (dashed) {x}={-0.47}
Vertex at {x,y} = {-0.47,-35.81}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-1.22, 0.00}
Root 2 at {x,y} = { 0.29, 0.00}
Solve Quadratic Equation by Completing The Square
4.2 Solving 63x2+59x-22 = 0 by Completing The Square .
Divide both sides of the equation by 63 to have 1 as the coefficient of the first term :
x2+(59/63)x-(22/63) = 0
Add 22/63 to both side of the equation :
x2+(59/63)x = 22/63
Now the clever bit: Take the coefficient of x , which is 59/63 , divide by two, giving 59/126 , and finally square it giving 3481/15876
Add 3481/15876 to both sides of the equation :
On the right hand side we have :
22/63 + 3481/15876 The common denominator of the two fractions is 15876 Adding (5544/15876)+(3481/15876) gives 9025/15876
So adding to both sides we finally get :
x2+(59/63)x+(3481/15876) = 9025/15876
Adding 3481/15876 has completed the left hand side into a perfect square :
x2+(59/63)x+(3481/15876) =
(x+(59/126)) • (x+(59/126)) =
(x+(59/126))2
Things which are equal to the same thing are also equal to one another. Since
x2+(59/63)x+(3481/15876) = 9025/15876 and
x2+(59/63)x+(3481/15876) = (x+(59/126))2
then, according to the law of transitivity,
(x+(59/126))2 = 9025/15876
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+(59/126))2 is
(x+(59/126))2/2 =
(x+(59/126))1 =
x+(59/126)
Now, applying the Square Root Principle to Eq. #4.2.1 we get:
x+(59/126) = √ 9025/15876
Subtract 59/126 from both sides to obtain:
x = -59/126 + √ 9025/15876
Since a square root has two values, one positive and the other negative
x2 + (59/63)x - (22/63) = 0
has two solutions:
x = -59/126 + √ 9025/15876
or
x = -59/126 - √ 9025/15876
Note that √ 9025/15876 can be written as
√ 9025 / √ 15876 which is 95 / 126
Solve Quadratic Equation using the Quadratic Formula
4.3 Solving 63x2+59x-22 = 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 = 63
B = 59
C = -22
Accordingly, B2 - 4AC =
3481 - (-5544) =
9025
Applying the quadratic formula :
-59 ± √ 9025
x = ———————
126
Can √ 9025 be simplified ?
Yes! The prime factorization of 9025 is
5•5•19•19
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).
√ 9025 = √ 5•5•19•19 =5•19•√ 1 =
± 95 • √ 1 =
± 95
So now we are looking at:
x = ( -59 ± 95) / 126
Two real solutions:
x =(-59+√9025)/126=(-59+95)/126= 0.286
or:
x =(-59-√9025)/126=(-59-95)/126= -1.222
Two solutions were found :
- x = -11/9 = -1.222
- x = 2/7 = 0.286
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