Question

solve (x-4)^2 - 5x -3 =0 and give your answer by correcting to 3 significant figures

Answer 1

$(x ^{2} - 8x + 16) - 5x - 3 = 0 \\ x ^{2} - 8x + 16 - 5x - 3 = 0 \\ {x}^{2} - 13x + 13 = 0$

Answer 2

Answer:

The solution is $x= 11.908$,$x= 1.092$

Step-by-step explanation:

Given: Equation

To find: solution of x and correct to 3 significant figures

Solution:

$\begin{aligned}&\left(x^{2}-8 x+16\right)-5 x-3=0 \\&x^{2}-8 x+16-5 x-3=0 \\&x^{2}-13 x+13=0\end{aligned}$

According to the Quadratic Formula,  x  , the solution for   $Ax^2+Bx+C = 0$, where  A, B  and  C  are numbers, often called coefficients, is given by :                            

$x=\frac{ -B\ ± \sqrt{B2-4AC} }{2A}$

 In our case,  $A = 1$ $B =-13$       $C = 13$                                

Accordingly, $B^2 - 4AC \\= 169 - 52 \\= 117$

Applying the quadratic formula :

              13 ± √ 117

  x  =    ——————

                     2

The prime factorization of  117   is$3*3*13$

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).

√ 117   =  √ 3•3•13   =

               ±  3 • √ 13

 √ 13   , rounded to 4 decimal digits, is   3.6056

So now we are looking at:

          x  =  ( 13 ± 3 •  3.606 ) / 2

Two real solutions:

x =(13+√117)/2=(13+3√ 13 )/2= 11.908

or:

x =(13-√117)/2=(13-3√ 13 )/2= 1.092