Question

Solve for x 0=3x^2+3x+7​

Answer 1

x =(3-√-75)/-6=1/-2+5i/6√ 3 = -0.5000-1.4434i

x =(3+√-75)/-6=1/-2-5i/6√ 3 = -0.5000+1.4434i

Explanation:

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

0-(3*x^2+3*x+7)=0

Step by step solution:

Step 1:

Equation at the end of step 1 :

0 - ((
`-3x^(2)` + 3x) + 7) = 0

Step 2:

Pulling out like terms:

2.1 Pull out like factors:

`-3x^(2)` - 3x - 7 = -1 • (
`3x^(2)` + 3x + 7)

Trying to factor by splitting the middle term

2.2 Factoring
`3x^(2)` + 3x + 7

The first term is,
`3x^(2)` its coefficient is 3 .

The middle term is, +3x its coefficient is 3 .

The last term, "the constant", is +7

Step-1 : Multiply the coefficient of the first term by the constant 3 • 7 = 21

Step-2 : Find two factors of 21 whose sum equals the coefficient of the middle term, which is 3 .

-21 + -1 = -22

-7 + -3 = -10

-3 + -7 = -10

-1 + -21 = -22

1 + 21 = 22

3 + 7 = 10

7 + 3 = 10

21 + 1 = 22

Observation : No two such factors can be found !!

Conclusion : Trinomial can not be factored

Equation at the end of step 3 :

`-3x^(2)` - 3x - 7 = 0

Step 3:

Parabola, Finding the Vertex:

3.1 Find the Vertex of y =
`-3x^(2)`-3x-7

For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is -0.5000

Plugging into the parabola formula -0.5000 for x we can calculate the y -coordinate :

y = -3.0 * -0.50 * -0.50 - 3.0 * -0.50 - 7.0

or y = -6.250

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y =
`-3x^(2)`-3x-7

Axis of Symmetry (dashed) {x}={-0.50}

Vertex at {x,y} = {-0.50,-6.25}

Function has no real roots

Solve Quadratic Equation by Completing The Square

3.2 Solving
`-3x^(2)`-3x-7 = 0 by Completing The Square .

Multiply both sides of the equation by (-1) to obtain positive coefficient for the first term:

`3x^(2)`+3x+7 = 0 Divide both sides of the equation by 3 to have 1 as the coefficient of the first term :

`x^(2)`+x+(7/3) = 0

Subtract 7/3 from both side of the equation :

`x^(2)`+x = -7/3

Now the clever bit: Take the coefficient of x , which is 1 , divide by two, giving 1/2 , and finally square it giving 1/4

Add 1/4 to both sides of the equation :

On the right hand side we have :

-7/3 + 1/4 The common denominator of the two fractions is 12 Adding (-28/12)+(3/12) gives -25/12

So adding to both sides we finally get :

`x^(2)`+x+(1/4) = -25/12

Adding 1/4 has completed the left hand side into a perfect square :

`x^(2)`+x+(1/4) =

(x+(1/2)) • (x+(1/2)) =

(x+(1/2))2

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

`x^(2)`+x+(1/4) = -25/12 and

`x^(2)`+x+(1/4) = (x+(1/2))2

then, according to the law of transitivity,

(x+(1/2))2 = -25/12

We'll refer to this Equation as Eq. #3.2.1

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

Note that the square root of

(x+(1/2))2 is

(x+(1/2))2/2 =

(x+(1/2))1 =

x+(1/2)

Now, applying the Square Root Principle to Eq. #4.2.1 we get:

x+(1/2) = √ -25/12

Subtract 1/2 from both sides to obtain:

x = -1/2 + √ -25/12

√ 3 , rounded to 4 decimal digits, is 1.7321

So now we are looking at:

x = ( 3 ± 5 • 1.732 i ) / -6

Two imaginary solutions :

x =(3+√-75)/-6=1/-2-5i/6√ 3 = -0.5000+1.4434i

or:

x =(3-√-75)/-6=1/-2+5i/6√ 3 = -0.5000-1.4434i