Solve Propertiesstraightline G(x)=4(x2+10x+25) - Tiger Algebra

Reformatting the input :

Changes made to your input should not affect the solution: (1): "x2" was replaced by "x^2".

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : g*(x)-(4*(x^2+10*x+25)-4*(25))=0

Step 1 :

Trying to factor by splitting the middle term

1.1 Factoring x2+10x+25 The first term is, x2 its coefficient is 1 .The middle term is, +10x its coefficient is 10 .The last term, "the constant", is +25 Step-1 : Multiply the coefficient of the first term by the constant 1 • 25 = 25 Step-2 : Find two factors of 25 whose sum equals the coefficient of the middle term, which is 10 .

-25	+	-1	=	-26
-5	+	-5	=	-10
-1	+	-25	=	-26
1	+	25	=	26
5	+	5	=	10	That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 5 and 5 x2 + 5x + 5x + 25Step-4 : Add up the first 2 terms, pulling out like factors : x • (x+5) Add up the last 2 terms, pulling out common factors : 5 • (x+5) Step-5 : Add up the four terms of step 4 : (x+5) • (x+5) Which is the desired factorization

Evaluate an expression :

1.2 Multiply (x+5) by (x+5) The rule says : To multiply exponential expressions which have the same base, add up their exponents.In our case, the common base is (x+5) and the exponents are : 1 , as (x+5) is the same number as (x+5)1 and 1 , as (x+5) is the same number as (x+5)1 The product is therefore, (x+5)(1+1) = (x+5)2

Equation at the end of step 1 :

gx - (4 • (x + 5)2 - 100) = 0

Step 2 :

Step 3 :

Pulling out like terms :

3.1 Pull out like factors : gx - 4x2 - 40x = x • (g - 4x - 40)

Equation at the end of step 3 :

x • (g - 4x - 40) = 0

Step 4 :

Theory - Roots of a product :

4.1 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 :

4.2 Solve : x = 0 Solution is x = 0

Equation of a Straight Line

4.3 Solve g-4x-40 = 0 Tiger recognizes that we have here an equation of a straight line. Such an equation is usually written y=mx+b ("y=mx+c" in the UK). "y=mx+b" is the formula of a straight line drawn on Cartesian coordinate system in which "y" is the vertical axis and "x" the horizontal axis.In this formula :y tells us how far up the line goesx tells us how far alongm is the Slope or Gradient i.e. how steep the line isb is the Y-intercept i.e. where the line crosses the Y axisThe X and Y intercepts and the Slope are called the line properties. We shall now graph the line g-4x-40 = 0 and calculate its properties

Graph of a Straight Line :

Calculate the Y-Intercept :

Notice that when x = 0 the value of g is 40/1 so this line "cuts" the g axis at g=40.00000

g-intercept = 40/1 = 40.00000

Calculate the X-Intercept :

When g = 0 the value of x is 10/-1 Our line therefore "cuts" the x axis at x=-10.00000

x-intercept = 40/-4 = 10/-1 = -10.00000

Calculate the Slope :

Slope is defined as the change in g divided by the change in x. We note that for x=0, the value of g is 40.000 and for x=2.000, the value of g is 48.000. So, for a change of 2.000 in x (The change in x is sometimes referred to as "RUN") we get a change of 48.000 - 40.000 = 8.000 in g. (The change in g is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

Slope = 8.000/2.000 = 4.000

Supplement : Solving Quadratic Equation Directly

Solving x2+10x+25 = 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 :

5.1 Find the Vertex of y = x2+10x+25Parabolas 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 -5.0000 Plugging into the parabola formula -5.0000 for x we can calculate the y -coordinate : y = 1.0 * -5.00 * -5.00 + 10.0 * -5.00 + 25.0 or y = 0.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = x2+10x+25 Vertex at {x,y} = {-5.00, 0.00} x-Intercept (Root) : One Root at {x,y}={-5.00, 0.00} Note that the root coincides with the Vertex and the Axis of Symmetry coinsides with the line x = 0

Solve Quadratic Equation by Completing The Square

5.2 Solving x2+10x+25 = 0 by Completing The Square . Subtract 25 from both side of the equation : x2+10x = -25Now the clever bit: Take the coefficient of x , which is 10 , divide by two, giving 5 , and finally square it giving 25 Add 25 to both sides of the equation : On the right hand side we have : -25 + 25 or, (-25/1)+(25/1) The common denominator of the two fractions is 1 Adding (-25/1)+(25/1) gives 0/1 So adding to both sides we finally get : x2+10x+25 = 0Adding 25 has completed the left hand side into a perfect square : x2+10x+25 = (x+5) • (x+5) = (x+5)2 Things which are equal to the same thing are also equal to one another. Since x2+10x+25 = 0 and x2+10x+25 = (x+5)2 then, according to the law of transitivity, (x+5)2 = 0We'll refer to this Equation as Eq. #5.2.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of (x+5)2 is (x+5)2/2 = (x+5)1 = x+5Now, applying the Square Root Principle to Eq. #5.2.1 we get: x+5 = √ 0 Subtract 5 from both sides to obtain: x = -5 + √ 0 The square root of zero is zero This quadratic equation has one solution only. That's because adding zero is the same as subtracting zero.The solution is: x = -5

Solve Quadratic Equation using the Quadratic Formula

5.3 Solving x2+10x+25 = 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 = 1 B = 10 C = 25 Accordingly, B2 - 4AC = 100 - 100 = 0Applying the quadratic formula : -10 ± √ 0 x = ————— 2The square root of zero is zero This quadratic equation has one solution only. That's because adding zero is the same as subtracting zero.The solution is: x = -10 / 2 = -5

Geometric figure: Straight Line

1. Slope = 8.000/2.000 = 4.000
2. x-intercept = 40/-4 = 10/-1 = -10.00000
3. g-intercept = 40/1 = 40.00000
4. x = 0