Question

x^2 - 10x + 21 = (x-7) (.............)​

Answer 1

Step-by-step explanation:

STEP

1

:

Trying to factor by splitting the middle term

1.1 Factoring x2-11x-14

The first term is, x2 its coefficient is 1 .

The middle term is, -11x its coefficient is -11 .

The last term, "the constant", is -14

Step-1 : Multiply the coefficient of the first term by the constant 1 • -14 = -14

Step-2 : Find two factors of -14 whose sum equals the coefficient of the middle term, which is -11 .

-14 + 1 = -13

-7 + 2 = -5

-2 + 7 = 5

-1 + 14 = 13

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Equation at the end of step

1

:

x2 - 11x - 14 = 0

STEP

2

:

Parabola, Finding the Vertex:

2.1 Find the Vertex of y = x2-11x-14

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, 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 5.5000

Plugging into the parabola formula 5.5000 for x we can calculate the y -coordinate :

y = 1.0 * 5.50 * 5.50 - 11.0 * 5.50 - 14.0

or y = -44.250

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = x2-11x-14

Axis of Symmetry (dashed) {x}={ 5.50}

Vertex at {x,y} = { 5.50,-44.25}

x -Intercepts (Roots) :

Root 1 at {x,y} = {-1.15, 0.00}

Root 2 at {x,y} = {12.15, 0.00}

Answer 2

Answer:

Compare with the quadratic equation which is

$a {x}^{2} + bx + c$

Here a = 1, b = -10 and c = 21

Now, Factorize the 21 which gives you 3×7.

Step2:

अब बीच वाली संख्या को ऐसे Break करना है कि जोड़ने या घटाने में 10 आये और गुणा करने पर 21 आए। अंक एक ही होने चाहिए।

हमने जो 21 के Factors निकाले हैं, उसमे यदि हम 3 और 7 को Add करें तो हमें 10 मिल जाएगा और यदि हम इन्ही अंकों का Multiple करें तो हमें 21 प्राप्त हो जाएगा।

हर संख्याओं में यह Apply होता है।

Step3:

${x}^{2} - 10x + 21 = {x}^{2} - 3x - 7x + 21$

${x}^{2} - 10x + 21 = {x}^{2} - 3x - 7x +21 \\ \\ \\ = {x}(x - 3) - 7(x - 3) \\ = (x - 7)(x - 3)$

Hence the R.H.S. answer will be = (x -3)