Math, asked by Tyeshsa5666, 2 months ago

What is the solution to the equation 45x−12x=310(x+4)?

Answers

Answered by anshu895252
1

Answer:

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Step-by-step explanation:

Step by Step Solution

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Step by step solution :

STEP

1

:

Equation at the end of step 1

((x4) - (32•5x2)) + 324 = 0

STEP

2

:

Trying to factor by splitting the middle term

2.1 Factoring x4-45x2+324

The first term is, x4 its coefficient is 1 .

The middle term is, -45x2 its coefficient is -45 .

The last term, "the constant", is +324

Step-1 : Multiply the coefficient of the first term by the constant 1 • 324 = 324

Step-2 : Find two factors of 324 whose sum equals the coefficient of the middle term, which is -45 .

-324 + -1 = -325

-162 + -2 = -164

-108 + -3 = -111

-81 + -4 = -85

-54 + -6 = -60

-36 + -9 = -45 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -36 and -9

x4 - 36x2 - 9x2 - 324

Step-4 : Add up the first 2 terms, pulling out like factors :

x2 • (x2-36)

Add up the last 2 terms, pulling out common factors :

9 • (x2-36)

Step-5 : Add up the four terms of step 4 :

(x2-9) • (x2-36)

Which is the desired factorization

Trying to factor as a Difference of Squares:

2.2 Factoring: x2-9

Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)

Proof : (A+B) • (A-B) =

A2 - AB + BA - B2 =

A2 - AB + AB - B2 =

A2 - B2

Note : AB = BA is the commutative property of multiplication.

Note : - AB + AB equals zero and is therefore eliminated from the expression.

Check : 9 is the square of 3

Check : x2 is the square of x1

Factorization is : (x + 3) • (x - 3)

Trying to factor as a Difference of Squares:

2.3 Factoring: x2 - 36

Check : 36 is the square of 6

Check : x2 is the square of x1

Factorization is : (x + 6) • (x - 6)

Equation at the end of step

2

:

(x + 3) • (x - 3) • (x + 6) • (x - 6) = 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 : x+3 = 0

Subtract 3 from both sides of the equation :

x = -3

Solving a Single Variable Equation:

3.3 Solve : x-3 = 0

Add 3 to both sides of the equation :

x = 3

Solving a Single Variable Equation:

3.4 Solve : x+6 = 0

Subtract 6 from both sides of the equation :

x = -6

Solving a Single Variable Equation:

3.5 Solve : x-6 = 0

Add 6 to both sides of the equation :

x = 6

Supplement : Solving Quadratic Equation Directly

Solving x4-45x2+324 = 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

Solving a Single Variable Equation:

Equations which are reducible to quadratic :

4.1 Solve x4-45x2+324 = 0

This equation is reducible to quadratic. What this means is that using a new variable, we can rewrite this equation as a quadratic equation Using w , such that w = x2 transforms the equation into :

w2-45w+324 = 0

Solving this new equation using the quadratic formula we get two real solutions :

36.0000 or 9.0000

Now that we know the value(s) of w , we can calculate x since x is √ w

Doing just this we discover that the solutions of

x4-45x2+324 = 0

are either :

x =√36.000 = 6.00000 or :

x =√36.000 = -6.00000 or :

x =√ 9.000 = 3.00000 or :

x =√ 9.000 = -3.00000

Four solutions were found :

x = 6

x = -6

x = 3

x = -3

Answered by aditya1222415364
1

Answer:

x = 1240/227

Step-by-step explanation:

45x - 12x = 310x + 1240

33 x = 310x + 1240

-310x + 33x = 1240

- 227x = 1240

x = 1240/227

hope it will helps you....

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