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- Quadratic Equations and Inequalities

Still Confused?

Try reviewing these fundamentals first

Still Confused?

Try reviewing these fundamentals first

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Get Started Now- Intro Lesson: a5:16
- Intro Lesson: b0:39
- Intro Lesson: c1:15
- Lesson: 1a1:25
- Lesson: 1b5:54
- Lesson: 2a1:27
- Lesson: 2b3:56
- Lesson: 310:39
- Lesson: 46:29
- Lesson: 5a5:17
- Lesson: 5b5:42

Basic Concepts: Solving linear equations using multiplication and division, Solving two-step linear equations: $ax + b = c$, ${x \over a} + b = c$, Solving literal equations, Power rule

In this lesson, we will learn:

- Solving Equations Using Direct Iteration
- Evaluating equations Using Iteration by Bisection
- Analyzing Equations Using Newton-Raphson Method

• 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:

- Introduction
__Introduction to solving polynomial equations by iteration__a)Direct/Fixed point iterationb)Iteration by bisectionc)Newton-Raphson method - 1.
**Solving Equations Using Direct Iteration**a)Show that $x^2-5x-8=0$ can be written in the form $x=\sqrt{8+5x}$.b)Use the iteration formula $x_{n+1}=\sqrt{8+5x_n}$ to find $x_3$ to $2$ decimal places. Start with $x_0=2$. - 2.
**Solving Equations Using Direct Iteration**a)Show that $x^3-x-8=0$ can be written in the form $x={^3}\sqrt{x+8}$.b)Use the iteration formula $x_{n+1}={^3}\sqrt{x_n+8}$ to find $x_4$ to $2$ decimal places. Start with $x_1=0$. - 3.
**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. - 4.Use bisection iteration to solve $x^3-x^2=39$. Give your answer to 1 decimal place.
- 5.
**Analyzing Equations Using Newton-Raphson Method**

Given $x^2-6x+5=0$.a)Find the iteration formula.b)Use the iteration formula found in (a) to approximate the solution. Start with $x_1=2$.

3.

Quadratic Equations and Inequalities

3.1

Solving quadratic equations by factoring

3.2

Solving quadratic equations by completing the square

3.3

Using quadratic formula to solve quadratic equations

3.4

Nature of roots of quadratic equations: The discriminant

3.5

Applications of quadratic equations

3.6

Solving quadratic inequalities

3.7

Solving polynomial equations by iteration