Question

Find the positive root of the equation x^3 + 2x^2 + 10x – 20 using Regula Falsi method and correct upto 4 decimal places​

Answer 1

x ≓ 1.368808150

2.1 x3+2x2+10x-20 is not a perfect cube

2.2 Factoring: x3+2x2+10x-20

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: 10x-20

Group 2: 2x2+x3

Pull out from each group separately :

Group 1: (x-2) • (10)

Group 2: (x+2) • (x2)

The groups have no common factor and can not be added up to form a multiplication.

2.3 Find roots (zeroes) of : F(x) = x3+2x2+10x-20

Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 1 and the Trailing Constant is -20.

The factor(s) are:

of the Leading Coefficient : 1

of the Trailing Constant : 1 ,2 ,4 ,5 ,10 ,20

We now use the Bisection Method to approximate one of the solutions. The Bisection Method is an iterative procedure to approximate a root (Root is another name for a solution of an equation).

The function is F(x) = x3 + 2x2 + 10x - 20

At x= 1.00 F(x) is equal to -7.00

At x= 2.00 F(x) is equal to 16.00

Intuitively we feel, and justly so, that since F(x) is negative on one side of the interval, and positive on the other side then, somewhere inside this interval, F(x) is zero

Procedure :

(1) Find a point "Left" where F(Left) < 0

(2) Find a point 'Right' where F(Right) > 0

(3) Compute 'Middle' the middle point of the interval [Left,Right]

(4) Calculate Value = F(Middle)

(5) If Value is close enough to zero goto Step (7)

Else :

If Value < 0 then : Left <- Middle

If Value > 0 then : Right <- Middle

(6) Loop back to Step (3)

(7) Done!! The approximation found is Middle

Follow Middle movements to understand how it works :

Next Middle will get us close enough to zero:

F( 1.368808150 ) is 0.000000896

The desired approximation of the solution is:

x ≓ 1.368808150

Answer 2

Step-by-step explanation:

Given:x³+2x²+10x-20=0

To find: find the real root correct to three significant figure by regula falsi method.

Solution:

Formula:

$\boxed{\bold{x = \frac{bf(a) - af(b)}{f(a) - f(b)}} }$

Step 1:Find a and b,put x=1

$f(1) = {(1)}^{3} + 2( {1)}^{2} + 10(1) - 20 \\ \\ f(1) = 1 + 2 + 10 - 20 \\ \\ \bold{\red{f(1) = - 7}}$

Thus, a=1 and f(a)= -7

put x=2

$f(2) = {(2)}^{3} + 2( {2)}^{2} + 10(2) - 20 \\ \\ f(2) = 8 + 8 + 20 - 20 \\ \\ \bold{\green{f(2) = 16}}$

Thus, b=2 and f(b)=16

There is a real root between x=1 and 2.

Step 2: Find next a,by putting value in formula

$x_1 = \frac{2( - 7) - 1(16)}{( - 7) - (16)} \\ \\ x_1 = \frac{ - 14 - 16}{ - 7 - 16} \\ \\ x_1 = \frac{ - 30}{ - 23} \\ \\ \bold{\pink{x_1 = 1.3043}}$

find the value of f(1.3043)

$f(1.3043) = {(1.3043)}^{3} + 2( {1.3043)}^{2} + 10(1.3043) - 20 \\ \\ \bold{\pink{f(1.3043) = - 1.3357}}$

Step 3:For next iteration a=1.3043 and f(a)=-1.3357

$x_2 = \frac{2( - 1.3357) - 1.3043(16)}{ - 1.3357- (16)} \\ \\ x_2 = \frac{ - 2.6714- 20.8688}{ - 17.3357} \\ \\ x_2 = \frac{-23.5402}{ - 17.3357} \\ \\ \bold{\green{x_2 = 1.3579}}$

$f(1.3579) = {(1.3579)}^{3} + 2( {1.3579)}^{2} + 10(1.3579) - 20 \\ \\ f(1.3579) = 2.5038 + 3.6877 + 13.579 - 20 \\ \\ \bold{\green{f(1.3579) = - 0.2295}}$

Step 4: For next iteration a=1.3579 and f(a)=-0.2295

$x_3= \frac{2( - 0.2295) - 1.3579(16)}{ - 0.2295 - 16} \\ \\ x_3 = \frac{ - 22.1854}{ - 16.2295} \\ \\ \bold{\purple{x_3 = 1.366}}$

Find f(1.366)

$f(1.3669) = {(1.3669)}^{3} + 2( {1.3669)}^{2} + 10(1.3669) - 20 \\ \\ \bold{\purple{f(1.3669) = - 0.0402}}$

Step 5: For next iteration a=1.3669 and f(a)=-0.0402

$x_4= \frac{2( - 0.0402) - 1.3669(16)}{ - 0.0402 - 16} \\ \\ x_4 = \frac{ - 21.9508}{ - 16.0402} \\ \\ \bold{\red{x_4= 1.3684}}$

*These steps can be iterate more to find more accurate root.

The real root exists more accurately at 1.3688

Final answer:

Real root of x³+2x²+10x-20=0 is approx 1.3688 upto 4 significant digits.

Hope it helps you.

To learn more on brainly:

Find the real root of the equation x³ -9x+1=0 by using regula falsi method.

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