Math, asked by datboi8, 7 months ago

Use the x-intertept method to find all real solutions of the equation.
-972 - 72x2 - 96x + 36 = 3x2 + x2 – 3x + 8

Answers

Answered by noahparmer0

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

(x+4) • (3x+7)

Which is the desired factorization

Equation at the end of step

(-3x - 7) • (x + 4) • (4x - 1) = 0

STEP

Theory - Roots of a product

6.1 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:

6.2 Solve : -3x-7 = 0

Add 7 to both sides of the equation :

-3x = 7

Multiply both sides of the equation by (-1) : 3x = -7

Divide both sides of the equation by 3:

x = -7/3 = -2.333

Solving a Single Variable Equation:

6.3 Solve : x+4 = 0

Subtract 4 from both sides of the equation :

x = -4

Solving a Single Variable bitggchyfd 4x = 1

Divide both sides of the equation by 4:

x = 1/4 = 0.250

Plugging into the parabola formula -3.1667 for x we can calculate the y -coordinate :

y = 3.0 * -3.17 * -3.17 + 19.0 * -3.17 + 28.0

or y = -2.083

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = 3x2+19x+28

Axis of Symmetry (dashed) {x}={-3.17}

Vertex at {x,y} = {-3.17,-2.08}

x -Intercepts (Roots) :

Root 1 at {x,y} = {-4.00, 0.00}

Root 2 at {x,y} = {-2.33, 0.00}

Solve Quadratic Equation by Completing The Square

7.2 Solving 3x2+19x+28 = 0 by Completing The Square .

Divide both sides of the equation by 3 to have 1 as the coefficient of the first term :

x2+(19/3)x+(28/3) = 0

Subtract 28/3 from both side of the equation :

x2+(19/3)x = -28/3

Now the clever bit: Take the coefficient of x , which is 19/3 , divide by two, giving 19/6 , and finally square it giving 361/36

Add 361/36 to both sides of the equation :

On the right hand side we have :

-28/3 + 361/36 The common denominator of the two fractions is 36 Adding (-336/36)+(361/36) gives 25/36

So adding to both sides we finally get :

x2+(19/3)x+(361/36) = 25/36

Adding 361/36 has completed the left hand side into a perfect square :

x2+(19/3)x+(361/36) =

(x+(19/6)) • (x+(19/6)) =

(x+(19/6))2

Things which are equal to the same thing are also equal to one another. Since

x2+(19/3)x+(361/36) = 25/36 and

x2+(19/3)x+(361/36) = (x+(19/6))2

then, according to the law of transitivity,

(x+(19/6))2 = 25/36

We'll refer to this Equation as Eq. #7.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of

(x+(19/6))2 is

(x+(19/6))2/2 =

(x+(19/6))1 =

x+(19/6)

Now, applying the Square Root Principle to Eq. #7.2.1 we get:

x+(19/6) = √ 25/36

Subtract 19/6 from both sides to obtain:

x = -19/6 + √ 25/36

Since a square root has two values, one positive and the other negative

x2 + (19/3)x + (28/3) = 0

has two solutions:

x = -19/6 + √ 25/36

x = -19/6 - √ 25/36

Note that √ 25/36 can be written as

√ 25 / √ 36 which is 5 / 6

Solve Quadratic Equation using the Quadratic Formula

7.3 Solving 3x2+19x+28 = 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 = ————————

In our case, A = 3

B = 19

C = 28

Accordingly, B2 - 4AC =

361 - 336 =

Applying the quadratic formula :

-19 ± √ 25

x = ——————

Can √ 25 be simplified ?

Yes! The prime factorization of 25 is

5•5

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).

√ 25 = √ 5•5 =

± 5 • √ 1 =

± 5

So now we are looking at:

x = ( -19 ± 5) / 6

Two real solutions:

x =(-19+√25)/6=(-19+5)/6= -2.333

or:

x =(-19-√25)/6=(-19-5)/6= -4.000

Three solutions were found :

x = 1/4 = 0.250

x = -4

x = -7/3 = -2.333

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Use the x-intertept method to find all real solutions of the equation.-972 - 72x2 - 96x + 36 = 3x2 + x2 – 3x + 8​

Answers

Use the x-intertept method to find all real solutions of the equation.
-972 - 72x2 - 96x + 36 = 3x2 + x2 – 3x + 8