Solving polynomial equations by iteration - Quadratic Equations

Solving polynomial equations by iteration

Lessons

Notes:

In this lesson, we will learn:

Solving Equations Using Direct Iteration

Evaluating equations Using Iteration by Bisection

Analyzing Equations Using Newton-Raphson Method

• Iteration means to repeatedly solving an equation to obtain a result using the result from the previous calculation.
• Direct iteration:

1. Rearrange the original equation such that the term in which the variable with the highest exponent is isolated.

2. Leave the variable on its own on the LHS by performing inverse operation.

3. The LHS becomes $x_{n+1}$ .

4. The RHS becomes $x_n$ .
• Iteration by bisection:

1. Shrink the interval where the roots lies within 2 equal parts.

2. Decide in which part the solution resides.

3. Repeat the steps until a consistent answer is achieved.
• Newton-Raphson method:
$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$

Intro Lesson

Introduction to solving polynomial equations by iteration
1.
Solving Equations Using Direct Iteration
2.
Solving Equations Using Direct Iteration
5.
Analyzing Equations Using Newton-Raphson Method
Given $x^2-6x+5=0$ .

Solving polynomial equations by iteration

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Solving linear equations using multiplication and division

Solving two-step linear equations: ax+b=cax + b = cax+b=c, xa+b=c{x \over a} + b = cax​+b=c

Solving literal equations

Power rule

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Don't just watch, practice makes perfect

Do better in math today

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Lessons

Notes:

Intro Lesson

Introduction to solving polynomial equations by iteration

a)

Direct/Fixed point iteration

b)

Iteration by bisection

c)

Newton-Raphson method

1.

Solving Equations Using Direct Iteration

a)

Show that x2−5x−8=0x^2-5x-8=0x2−5x−8=0 can be written in the form x=8+5xx=\sqrt{8+5x}x=8+5x​.

b)

Use the iteration formula xn+1=8+5xnx_{n+1}=\sqrt{8+5x_n}xn+1​=8+5xn​​ to find x3x_3x3​ to 222 decimal places. Start with x0=2x_0=2x0​=2.

2.

Solving Equations Using Direct Iteration

a)

Show that x3−x−8=0x^3-x-8=0x3−x−8=0 can be written in the form x=3x+8x={^3}\sqrt{x+8}x=3x+8​.

b)

Use the iteration formula xn+1=3xn+8x_{n+1}={^3}\sqrt{x_n+8}xn+1​=3xn​+8​ to find x4x_4x4​ to 222 decimal places. Start with x1=0x_1=0x1​=0.

3.

Evaluating equations Using Iteration by Bisection The equation x3+5x−7=91x^3+5x-7=91x3+5x−7=91 has a solution between 4 and 5. Use bisection iteration to find the solution and give the answer to 1 decimal place.

4.

Use bisection iteration to solve x3−x2=39x^3-x^2=39x3−x2=39. Give your answer to 1 decimal place.

5.

Analyzing Equations Using Newton-Raphson Method Given x2−6x+5=0x^2-6x+5=0x2−6x+5=0.

a)

Find the iteration formula.

b)

Use the iteration formula found in (a) to approximate the solution. Start with x1=2x_1=2x1​=2.

Solving polynomial equations by iteration

Don't just watch, practice makes perfect.

We have over 1850 practice questions in Algebra for you to master.

Algebra
Solving linear equations using multiplication and division

Algebra
Solving two-step linear equations: $ax + b = c$ , ${x \over a} + b = c$

Algebra
Solving literal equations

Calculus
Power rule

Solving two-step linear equations: $ax + b = c$ , ${x \over a} + b = c$

or

Show that $x^2-5x-8=0$ can be written in the form $x=\sqrt{8+5x}$ .

Use the iteration formula $x_{n+1}=\sqrt{8+5x_n}$ to find $x_3$ to $2$ decimal places. Start with $x_0=2$ .

Show that $x^3-x-8=0$ can be written in the form $x={^3}\sqrt{x+8}$ .

Use the iteration formula $x_{n+1}={^3}\sqrt{x_n+8}$ to find $x_4$ to $2$ decimal places. Start with $x_1=0$ .

Evaluating equations Using Iteration by Bisection
The equation $x^3+5x-7=91$ has a solution between 4 and 5. Use bisection iteration to find the solution and give the answer to 1 decimal place.

Use bisection iteration to solve $x^3-x^2=39$ . Give your answer to 1 decimal place.

Analyzing Equations Using Newton-Raphson Method
Given $x^2-6x+5=0$ .

Use the iteration formula found in (a) to approximate the solution. Start with $x_1=2$ .