Question

find the roots of 4xsquare+ 3x+5=0 by the method of completing the square.​

Answer 1

Step-by-step explanation:

Hope these will help you

Answer 2

hi mate,

solution :

Solving 4x²+3x+5 = 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 = ————————

2A

In our case, A = 4

B = 3

C = 5

Accordingly, B² - 4AC = 9 - 80 = -71

Applying the quadratic formula :

-3 ± √ -71

x = ——————

8

In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written (a+b*i)

Both i and -i are the square roots of minus 1

Accordingly,√ -71 =

√ 71 • (-1) =

√ 71 • √ -1 =

± √ 71 • i

√ 71 , rounded to 4 decimal digits, is 8.4261

So now we are looking at:

x = ( -3 ± 8.426 i ) / 8

Two imaginary solutions :

x =(-3+√-71)/8 = (-3+i√ 71 )/8 =

-0.3750+1.0533i

or:

x =(-3-√-71)/8=(-3-i√ 71 )/8=

-0.3750-1.0533i

Two solutions were found :

x = (-3-√-71)/8=(-3-i√ 71 )/8

= - 0.3750-1.0533i

x =(-3+√-71)/8=(-3+i√71 )8

= - 0.3750+1.0533i

Two solutions were found :

x =(-3-√-71)/8=(-3-i√ 71 )/8

= -0.3750-1.0533i

x =(-3+√-71)/8=(-3+i√ 71 )/8

= -0.3750+1.0533i

or..

Solving 4x2+3x+5 = 0 by Completing The Square .

Divide both sides of the equation by 4 to have 1 as the coefficient of the first term :

x2+(3/4)x+(5/4) = 0

Subtract 5/4 from both side of the equation :

x2+(3/4)x = -5/4

Now the clever bit: Take the coefficient of x , which is 3/4 , divide by two, giving 3/8 , and finally square it giving 9/64

Add 9/64 to both sides of the equation :

On the right hand side we have :

-5/4 + 9/64 The common denominator of the two fractions is 64 Adding (-80/64)+(9/64) gives -71/64

So adding to both sides we finally get :

x2+(3/4)x+(9/64) = -71/64

Adding 9/64 has completed the left hand side into a perfect square :

x2+(3/4)x+(9/64) =

(x+(3/8)) • (x+(3/8)) =

(x+(3/8))2

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

x2+(3/4)x+(9/64) = -71/64 and

x2+(3/4)x+(9/64) = (x+(3/8))2

then, according to the law of transitivity,

(x+(3/8))2 = -71/64

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+(3/8))2 is

(x+(3/8))2/2 =

(x+(3/8))1 =

x+(3/8)

Now, applying the Square Root Principle to Eq. #3.2.1 we get:

x+(3/8) = √ -71/64

Subtract 3/8 from both sides to obtain:

x = -3/8 + √ -71/64

In Math, i is called the imaginary unit. It satisfies i2 =-1. Both i and -i are the square roots of -1

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

x2 + (3/4)x + (5/4) = 0

has two solutions:

x = -3/8 + √ 71/64 • i

or

x = -3/8 - √ 71/64 • i

Note that √ 71/64 can be written as

√ 71 / √ 64 which is √ 71 / 8

i hope it helps you.