Solving Equations

Equation of degree1Equation of degree2

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Introduction- Objectives

Solving equations can get confusing when too many variables are floating about. But when you keep thing straight it is easy and fun

Lesson Objectives: From the presented materials ,at the end of the lesson, students should be able to solve any equation of fist and second.

Outcomes•At the end of this lesson, all students should be able to: Understand how the mechanism of equation of degree1•At the end of this lesson, all students should be able to: Understand how the mechanism of equation of degree2•At the end of this lesson, all students should be able to: Solve with confident any equations of degree 1 and 2

Equations of degree1A first-degree equation is called a linear equation. The highest exponent of a linear equation is 1.

The standard form for a linear equation is: ax + b = c, where a, b, and c are constants (numbers).

To solve a first-degree equation, we use the following steps:

• If parentheses occur, multiply to remove them using the parentheses rules learned.

• Collect like terms.

• Use the addition/subtraction property to get all terms with a variable on one side and all numbers

on the other side.

• Collect like terms.

• Apply the multiplication/division property to solve for the variable.

• Verify the solution.

Remember:

Equations behave like a balance. So we need to apply the same operation to both sides of an

equation to maintain the balance. This means we can:

• add the same number to both sides of an equation

• subtract the same number from both sides of an equation

• multiply both sides of an equation by the same number

• divide both sides of an equation by the same number

Example14x-16=0

In solving this equation we want find the value of X that make the expression true. “X” is then called the unknown.

When dealing with equations it is better to place all known value on one side of the “=” sign.

But remember everything you move changes its sign. If the sign was “+” it changes to “– “and vice versa

So: we can separate 4x and 16

4x=16. The sign of 16 changes from -16 to +16

Find the value of X.

To find X all we have to do is divide 16 by 4. So x=4

Let’s verify:

4x4=16

The general formula of what we just did is :

Ax+b=o

Then X=b/a

4X+8 = -12

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X = 4 X = 5 X = -5

1. X = 4

2. X = 5

3. X = -5

Example2

5X+2 = -X+10

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33% 33%33%1. X = 2

2. X = 4/3

3. X = -7

Example3

Y/10 = 2

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Y = 4 Y = 20 Y = -5

1. Y = 4

2. Y = 20

3. Y = -5

Example4 With parentheses

3(2X-3)+4X =5(X+4)-9

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33% 33%33%1. X = 4

2. X =12

3. X =1

Equations of degree2• First degree equations

• A second-degree equation is called a quadratic equation. The highest exponent of a quadratic equation is 2.The general form of these equations is ax2 +bx +c = 0; where a, b, and c are constants and “a” is not equal 0.

• The solution for this type of equation can often be found by a method known as factoring, because really the second degree equation is the product of two first degree equations. so it can be factored into these equations.

• BUT, Finding the factors of a quadratic equation is not always easy. To solve this problem, the quadratic formula was invented so that any quadratic equation can be solved. The quadratic equation is stated as follows for the general equation ax2 + bx + c = 0

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REMEMBER:

The second degree equation has no real number solution if 𝑏2 − 4𝑎𝑐< 0

Example1

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Example1: Solve for X 𝑥2 − 3𝑋+ 2 = 0

Here a=1, b=-3, c=2

So, 𝑥= −(−3)±ඥ(−3)2−4.1.22.1

X=2 with the plus sign

X=1 with the minus sign

x2 - x – 6 =0

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33%1. X=(1,4)

2. X=(5,8)

3. X=(-2,3)

Example2

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Example2: solve for X 2𝑥2 − 3𝑥+ 1 = 0

A=2, b=-3, c=1, 𝑏2 − 4𝑎𝑐= 1

So according to the quadratic formula

X=1 using the plus sign

X=12 using the minus sign

x2 + 4x +4 =0

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X=(3,6) X=-2 X=(1,0)

1. X=(3,6)

2. X=-2

3. X=(1,0)

Example3

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Example3 𝑥2 − 9 = 0

Here, a=1, b = 0, c = -9

Instead using the quadratic formula, we can apply the method we used in the first degree equation, add 9 to both sides.

We get 𝑥2 = 9

X=-3 or 3

x2 - 16 =0

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33%

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33%1. X=(2,2)

2. X=(0,1)

3. X=(-4,4)

Example: Using factors

If you can easily find the factors the composed the second degree equation you

can use it instead of the quadratic formula.   

• x2 - 1 = 0

by factoring we get

• (x + 1)(x - 1) = 0

• x = ±1

Or this equations

• x2 - 5x + 6 = 0

• (x - 2)(x - 3) = 0

• x = 2, 3

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(X-6)(X+5)=0

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1. X=(0,0)

2. X=(6,-5)

3. X=(7,4)

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