LINEAR AND QUADRATIC EQUATIONS

Quant - Linear and Quadratic Equations

What are the 2 methods for solving Linear Equations in 2 variables?

• Substitution Method:

  • Isolate one of the variables in either equation

  • Insert the value of that variable in other equation

• Combination Method:

  • Add or subtract one equation from/into another equation to eliminate one variable and solve for the other variable

How to deal with equations which contain fractions?

Eliminate those fractions by multiplying each term by the LCM

What’s an LCM?

LCM is the smallest non zero whole number that is divisible by each of the numbers

If the product of 2 integers is 1, what can the two integers be?

Both integers can either be 1 or -1

What is the zero product property?

If the product of 2 quantities is 0, at least one of the quantities has to be equal to 0

What is the general form of a quadratic equation?

ax² + bx + c = 0

What is the formula for finding the roots/factors of an equation?

x = (-b ± √ (b² - 4ac) )/2a

(x+1) and (x+8) vs x = -1 and x = -8?
What is each called

• (x+1) and (x+8) are called the factors of the quadratic equation

• X = -1 and x = -8 are the solutions or roots of the quadratic equation

What is an equation and how is it different to an entity?

• An equation is a statement that is truly for only one or some values of the variable(s). For eg: 2x-3=5

• An identity is one that is true for all infinitely many values of the variable(s). For eg: x+x=2x or x²/x = x

(x+y)² = ?
(x-y)² = ?

(x+y)(x-y) = ?

• (x+y)² = x ² + y² + 2xy

• (x-y)² = x² + y² - 2xy

• (x+y)(x-y) = x²-y²

Is it necessary that 2 equations are sufficient to determine the values of 2 variables?

No. Sometimes two equations appear different but are actually the same

Can we determine the values of a greater number of variables using fewer unique equations containing those variables?

Yes. We can sometimes solve a single equation containing 2 variables

Can we divide by a variable when we don’t know whether it’s equal to 0?

No. We can only divide by a variable when we know that the variable is not equal to 0.

How do we determine the number of roots of a quadratic equation ax² + bx + c = 0?

For the quadratic equation ax² + bx + c = 0, the discriminant is b² - 4ac

Value of discriminant b² - 4ac	Number of roots/solutions
+ve	2
0	1
-ve	0

How do you calculate the sum and product of the quadratic equation ax² + bx + c = 0 without finding out the roots (doing the full calculation)?

• Sum = -b/a

• Product = c/a