Algebra

You have learned some basics of algebra at study plan for beginners, now you can move on to in-depth analysis of numbers and their properties. Remember that algebra is the key to success, as I said in study plan for beginners as well. So never skip this topic in any way even if you don’t like math at all.

Exponents and roots:

Remember the following trend in a number line which is split into 4 parts:

This figure gives information about change (whether increase or decrease), when we take square, square root, cube and cube root of x, where x is any number, that fall under the mentioned category in a split number link above.

From this figure, you may extract information as bellow:

If x² > x³

This means either x lies between 0 and 1 or x is a negative number. Because how it’s possible that a square in power/exponent of a number is greater than a cube in the power/exponent of that number. It’s possible only when x is less than 1, but not zero.

You may also extract the following thing from the important figure above:

1. In number-line, when a number greater than 1 is selected, then it’s cube (x³) is greater than its square (x²). But square root of x is greater than cube root of x. In short, increasing the power of x would increase it’s value vice versa.

2. While, when a number, less than 1 but greater than 0, is selected from the number-line, it’s behavior is exactly opposite to the number that is greater than 1 as discuss in previous sentence. In short, increasing the power of x would decrease it’s value, and vice versa.

3. When a number, less than 0 but greater than -1, is selected, it’s behavior you may also see from the figure how it fluctuate in different exponential values. In short, when exponent is even, value would become positive and increase by greater amount then when taking odd power.

4. When a number, less than -1 is selected, is selected, it’s behavior is always in a way that when it has an even integer in its power, it would become positive, otherwise it would remain negative.

This means you must be aware whether the number is from which category when solving exponential inequalities. For instance,

If x² > x³, which of the following MUST be true?

I. x > x²
II. x² > x
III. √x > x

A) I only
B) II only
C) III only
D) I and II only
E) None of the above

What you think about the answer?

Clearly its none of these. because either x may lie between 0 and 1 or x is negative. Option I would be false if x is negative, II is false if x is lie between 0 and 1 and option III is false if x is negative, because square root of negative is undefined. So we cannot say undefined is greater than x.

Solving Algebraic Expressions:
As you already know the exponents & roots. Let’s learn how to simplify those expression.

x² = 16
⇒ x = ±4

So here, x = 4     or     x = –4

But if,

y³ = 125
⇒ y = 5

So we learned that if any variable raised to power even equals to an integer, then there would be two possible answers of that variable. While if the variable raised to power odd equals to an integer, then there would be only one possible positive answer of that variable.

There are important algebraic rules that you have learned in beginners study plan; Just remember those.

And remember that:

√16 = +4 only

I’m sure you might confuse with this thing. i.e why

√16 = +4

While,

x² = 16

⇒ x = √16

⇒ x = ±4

Then why √16 ≠ ±4   but only +4?

Remember that x is a variable, while √16 is a constant. A constant only has one value, while a variable might have all possible value that can equate the equation.

For instance,

If you check back the equation by putting the value of x in it, the equation will satisfy by putting both of the values of x (i.e +4 and –4).

Also it is very important to remember that in any quadratic equation, always check back both of the values of the variable in order to see whether the equation is satisfy? For instance, if the the quadratic equation is given such that the variable x is inside square root. If after solution, you get a positive and a negative value of variable x, then never consider negative value of variable x, because answer of square root is always positive or minimum 0. To understand this let’s consider the following scenario of quadratic equation that we have learned in beginners study plan:

Solve for value of x:

√3x² + 10x + 24 = 2x ———————– (eq. 1)

Remember that we solve such expression by taking square on both sides to remove square root:

(√3x² + 10x + 24)² = (2x)²

⇒ 3x² + 10x + 24 = 4x²

⇒ x² – 10x – 24 = 0

Now, by factorizing,

⇒ x² – 12x + 2x – 24 = 0                   (As sum of –12x and 2x is –10x; and product of –12 and 2 is –24)

⇒ x² – 12x + 2x – 24 = 0

⇒ x(x – 12) + 2(x – 12) = 0

⇒ (x + 2) (x – 12) = 0

⇒ Either     x = –2         or         x = 12

But when you see (eq.1), you must start believing that x cannot be negative, otherwise the right hand side of (eq.1) will become negative. As you know the left hand side is a square root of something, which cannot be negative. Therefore,

⇒ x ≠ –2

and

⇒ x = 12 Answer

Now, let’s solve a complex problem with smart work approach.

Find value of x, if

(√3x² + 10x + 24)² + 5√3x² + 10x + 24 – 6 = 0

Simply suppose:

A = √3x² + 10x + 24

A² + 5A – 6 = 0

By factorizing,

⇒ A² + 6A – A – 6 = 0

⇒ A(A + 6) – 1(A + 6) = 0

⇒ Either   (A – 1) = 0         or         (A + 6) = 0

⇒ A = 1         or         A = –6

But, as A is a square root of something, therefore A cannot be negative; Thus,

A = 1

⇒ √3x² + 10x + 24 = 1

Now, you can solve it for x by taking square on both sides as we’ve learned previously. And again , if you get two values of x such that by putting one of the value does not satisfy the equation, then there will be only one answer, otherwise two answers.

Also remember that if,

√x² – 10 = 1

And let’s suppose (just suppose), after solving one of the values of x comes out to be = 2   or   –2

Now, if you put these values value to original equation (i.e √x² – 10 = 1) it will not satisfy because the left hand side of the equation is undefined. Because square root of a negative value is undefined.

These questions are usually tested with questions wording like: ‘which of the following could be or cannot be the value of x?’ etc…

Equation of Line:
Equation of line is descried as:
y = mx + c

Where,
y → y coordinate
m → slope of the line
x → x coordinate
c → y intercept

When slop is negative, it means the link is on higher points on left side than the right side of coordinate plane, as shown below:

Here you see than the line at right side of vertical axis is higher than the line at left side of vertical axis. It has negative slop.

On other hand if the line on the right side of vertical axis is higher than the left side of vertical axis, then the slop is positive.

Suppose,

y = ^5x⁄₄ + 5

Here,
slop = m = ^5x⁄₄
y-intercept = c = 5

Remember: y-intercept is the point where the line intersect on y-axis (i.e vertical axis)

Now you may draw this easily as bellow:

put x = 0 on the equation above, you’ll get y-intercept as 5. And put y = 0, you’ll get x-intercept as –4. Just joint these two points from the coordinate plane, you’ll get the line in coordinate plane as below: