Q:

solve for x 0=3x^2+3x+7​

Accepted Solution

A:
Answer: x =(3-√-75)/-6=1/-2+5i/6√ 3 = -0.5000-1.4434i x =(3+√-75)/-6=1/-2-5i/6√ 3 = -0.5000+1.4434iStep-by-step 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 -  (([tex]-3x^{2}[/tex] +  3x) +  7)  = 0  Step  2:Pulling out like terms: 2.1     Pull out like factors:   [tex]-3x^{2}[/tex] - 3x - 7  =   -1 • ([tex]3x^{2}[/tex] + 3x + 7) Trying to factor by splitting the middle term 2.2     Factoring  [tex]3x^{2}[/tex] + 3x + 7 The first term is,  [tex]3x^{2}[/tex]  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 factoredEquation at the end of step  3  :  [tex]-3x^{2}[/tex] - 3x - 7  = 0 Step  3:Parabola, Finding the Vertex: 3.1      Find the Vertex of   y = [tex]-3x^{2}[/tex]-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 - 7.0or   y = -6.250Parabola, Graphing Vertex and X-Intercepts :Root plot for :  y = [tex]-3x^{2}[/tex]-3x-7Axis of Symmetry (dashed)  {x}={-0.50} Vertex at  {x,y} = {-0.50,-6.25} Function has no real rootsSolve Quadratic Equation by Completing The Square 3.2     Solving   [tex]-3x^{2}[/tex]-3x-7 = 0 by Completing The Square . Multiply both sides of the equation by  (-1)  to obtain positive coefficient for the first term: [tex]3x^{2}[/tex]+3x+7 = 0  Divide both sides of the equation by  3  to have 1 as the coefficient of the first term :   [tex]x^{2}[/tex]+x+(7/3) = 0Subtract  7/3  from both side of the equation :   [tex]x^{2}[/tex]+x = -7/3Now 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 :   [tex]x^{2}[/tex]+x+(1/4) = -25/12Adding  1/4  has completed the left hand side into a perfect square :   [tex]x^{2}[/tex]+x+(1/4)  =   (x+(1/2)) • (x+(1/2))  =  (x+(1/2))2Things which are equal to the same thing are also equal to one another. Since   [tex]x^{2}[/tex]+x+(1/4) = -25/12 and   [tex]x^{2}[/tex]+x+(1/4) = (x+(1/2))2then, according to the law of transitivity,   (x+(1/2))2 = -25/12We'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) = √ -25/12Subtract  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 ) / -6Two 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