# If (x + 1)(x + 2)(x + 3)(x + 6) = 3x^2, then the equation has

### Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : (x+1)*(x+2)*(x+3)*(x+6)-(168*x^2)=0

## Step 1 :

Equation at the over of step 1 :

((((x+1)•(x+2))•(x+3))•(x+6))-(23•3•7x2) = 0

## Step 2 :

Equation at the kết thúc of step 2 : (((x+1)•(x+2)•(x+3))•(x+6))-(23•3•7x2) = 0

## Step 3 :

Equation at the over of step 3 : ((x+1)•(x+2)•(x+3)•(x+6))-(23•3•7x2) = 0

## Step 4 :

Equation at the end of step 4 : (x+1)•(x+2)•(x+3)•(x+6)-(23•3•7x2) = 0

## Step 5 :

### Polynomial Roots Calculator :

5.1 Find roots (zeroes) of : F(x) = x4+12x3-121x2+72x+36Polynomial Roots Calculator is a set of methods aimed at finding values ofxfor which F(x)=0 Rational Roots thử nghiệm is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integersThe Rational Root Theorem states that if a polynomial zeroes for a rational numberP/Q then p is a factor of the Trailing Constant and Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 1 and the Trailing Constant is 36. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,9 ,12 ,18 ,36 Let us kiểm tra ....

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PQP/QF(P/Q)Divisor
-11 -1.00 -168.00
-21 -2.00 -672.00
-31 -3.00-1512.00
-41 -4.00-2700.00
-61 -6.00-6048.00
-91 -9.00-12600.00
-121-12.00-18252.00
-181-18.00-5472.00
-361-36.00960372.00
11 1.00 0.00x-1
21 2.00 -192.00
31 3.00 -432.00
41 4.00 -588.00
61 6.00 0.00x-6
91 9.00 6192.00
121 12.0024948.00
181 18.00137088.00
361 36.002085300.00

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p chú ý that q and p originate from P/Q reduced khổng lồ its lowest terms In our case this means that x4+12x3-121x2+72x+36can be divided by 2 different polynomials,including by x-6

### Polynomial Long Division :

5.2 Polynomial Long Division Dividing : x4+12x3-121x2+72x+36("Dividend") By:x-6("Divisor")

 dividend x4 + 12x3 - 121x2 + 72x + 36 -divisor * x3 x4 - 6x3 remainder 18x3 - 121x2 + 72x + 36 -divisor * 18x2 18x3 - 108x2 remainder - 13x2 + 72x + 36 -divisor * -13x1 - 13x2 + 78x remainder - 6x + 36 -divisor * -6x0 - 6x + 36 remainder 0

Quotient : x3+18x2-13x-6 Remainder: 0

### Polynomial Roots Calculator :

5.3 Find roots (zeroes) of : F(x) = x3+18x2-13x-6See theory in step 5.1 In this case, the Leading Coefficient is 1 and the Trailing Constant is -6. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 ,2 ,3 ,6 Let us thử nghiệm ....

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PQP/QF(P/Q)Divisor
-11 -1.00 24.00
-21 -2.00 84.00
-31 -3.00 168.00
-61 -6.00 504.00
11 1.00 0.00x-1
21 2.00 48.00
31 3.00 144.00
61 6.00 780.00

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p lưu ý that q and phường originate from P/Q reduced to its lowest terms In our case this means that x3+18x2-13x-6can be divided with x-1

### Polynomial Long Division :

5.4 Polynomial Long Division Dividing : x3+18x2-13x-6("Dividend") By:x-1("Divisor")

 dividend x3 + 18x2 - 13x - 6 -divisor * x2 x3 - x2 remainder 19x2 - 13x - 6 -divisor * 19x1 19x2 - 19x remainder 6x - 6 -divisor * 6x0 6x - 6 remainder 0

Quotient : x2+19x+6 Remainder: 0

Trying to factor by splitting the middle term

5.5Factoring x2+19x+6 The first term is, x2 its coefficient is 1.The middle term is, +19x its coefficient is 19.The last term, "the constant", is +6Step-1 : Multiply the coefficient of the first term by the constant 1•6=6Step-2 : Find two factors of 6 whose sum equals the coefficient of the middle term, which is 19.

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 -6 + -1 = -7 -3 + -2 = -5 -2 + -3 = -5 -1 + -6 = -7 1 + 6 = 7 2 + 3 = 5 3 + 2 = 5 6 + 1 = 7

Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored

Equation at the kết thúc of step 5 :

(x2 + 19x + 6) • (x - 1) • (x - 6) = 0

## Step 6 :

Theory - Roots of a sản phẩm :6.1 A product of several terms equals zero.When a hàng hóa 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 lớn solve as many equations as there are terms in the productAny solution of term = 0 solves hàng hóa = 0 as well.

Parabola, Finding the Vertex:6.2Find the Vertex ofy = x2+19x+6Parabolas 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 khổng lồ 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 -9.5000Plugging into the parabola formula -9.5000 for x we can calculate the y-coordinate:y = 1.0 * -9.50 * -9.50 + 19.0 * -9.50 + 6.0 or y = -84.250

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = x2+19x+6 Axis of Symmetry (dashed) x=-9.50 Vertex at x,y = -9.50,-84.25 x-Intercepts (Roots) : Root 1 at x,y = -18.68, 0.00 Root 2 at x,y = -0.32, 0.00

Solve Quadratic Equation by Completing The Square

6.3Solvingx2+19x+6 = 0 by Completing The Square.Subtract 6 from both side of the equation :x2+19x = -6Now the clever bit: Take the coefficient of x, which is 19, divide by two, giving 19/2, and finally square it giving 361/4Add 361/4 to lớn both sides of the equation :On the right hand side we have:-6+361/4or, (-6/1)+(361/4)The common denominator of the two fractions is 4Adding (-24/4)+(361/4) gives 337/4So adding lớn both sides we finally get:x2+19x+(361/4) = 337/4Adding 361/4 has completed the left hand side into a perfect square :x2+19x+(361/4)=(x+(19/2))•(x+(19/2))=(x+(19/2))2 Things which are equal to lớn the same thing are also equal to one another. Sincex2+19x+(361/4) = 337/4 andx2+19x+(361/4) = (x+(19/2))2 then, according lớn the law of transitivity,(x+(19/2))2 = 337/4We"ll refer to lớn this Equation as Eq. #6.3.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x+(19/2))2 is(x+(19/2))2/2=(x+(19/2))1=x+(19/2)Now, applying the Square Root Principle to Eq.#6.3.1 we get:x+(19/2)= √ 337/4 Subtract 19/2 from both sides to obtain:x = -19/2 + √ 337/4 Since a square root has two values, one positive and the other negativex2 + 19x + 6 = 0has two solutions:x = -19/2 + √ 337/4 orx = -19/2 - √ 337/4 chú ý that √ 337/4 can be written as√337 / √4which is √337 / 2