Solving Quadratic Roots

In mathematics, linear inequalities are defined as those linear equations that have inequality symbols. In simpler words, linear inequalities in mathematical expressions are those inequalities that do not completely satisfy the equation when represented in an expression. The symbols used to represent linear inequalities are <, >, ≥, and ≤. 

A quadratic equation is one such equation that in a standard form can be expressed in ax2+bx+c = 0, where x is a variable and ‘a’ does not equal 0. The roots of a quadratic equation are defined as those possible values of the variable ‘x’ that satisfies the equation.

Linear Inequalities 

Linear inequalities are defined as those mathematical expressions that do not entirely satisfy the equation. To understand the concept of linear inequality, let us look at an example.

Let’s say you have 100 rupees with you and you want to buy some pens with it. Consider one pen costs you 4 rupees and eight paise (4.8 rupees). If you buy ‘x’ number of pens, it is safe to say that you spent 4.8x rupees on your purchase. However, since we know that the value of x will always be in absolute numbers, the total money spent can never be a hundred or more than a hundred.

Therefore, in mathematical expressions, this purchase can be expressed in the following way:

4.80 x < 100 .….. (i)

Now let us look at a different scenario. Imagine a man who wants to buy some pillows and blankets. Considering this man has a budget of 3,000 rupees with him, now, let’s say that a pillow costs him 300 and a blanket costs him 600 rupees. If the man buys x number of pillows and y number of blankets, this transaction can be expressed in the following manner:

300x + 600y ≤ 3,000

or, x + 2y ≤ 1000 ……(ii)

Looking at both of these examples, we can see that equation (i) has a < sign and equation (ii) has ≤ sign; in it, these signs represent inequalities of the equation.

Particularly looking at equation (ii), the equation can have two explanations, that is,

 x + 2y < 1000, and x + 2y = 1000. In these expressions, the former represents linear inequality.

Such linear equations that do not give a definite equal value to the equation are called linear inequalities.

Quadratic Equations

Quadratic equations are those mathematical expressions of equations that, in a standard form, can be expressed in the following manner:

ax2 + bx + c = 0

Here, a, b, and c are constants, while ‘x’ is a variable. In quadratic equations, the value of ‘a’ can never be 0; otherwise, the equation becomes a linear expression. Therefore, the ‘ax2‘ component of the equation makes it the quadratic equation.

Roots of the Quadratic Equation

The roots of the quadratic equations are defined as those values of the variable ‘x’ that satisfy the entire equation.

Let us look at one example.

Consider a quadratic equation:

x2 + 2x – 8 = 0

Let us prove that 2 is the solution of the quadratic equation. Now substituting the variable x with 2 in the equation gives

(2)2 + 2(2) – 8 = 0,

We find that, L.H.S = R.H.S.

Here, value 2 satisfies the equation, and therefore, 2 is a root of the equation.

Now, check it for x = – 4

Substituting the variable x with -4:

(-4)2 + 2(-4)-8 = 0,

16-8-8= 0

Again, we find L.H.S = R.H.S

-4 also satisfies the equation; therefore, -4 is the root of the equation.

Finally, consider 3 for the value of x:

The equation becomes

(3)2 + 2(3)- 8 = 0

9 + 6 – 8 = 0

We find, L.H.S ≠ R.H.S. Hence, 3 cannot be the root of the equation.

Therefore, for the above equation, there can only be two roots, 2 and -4

If there is a quadratic equation, ax2 + bx + c = 0, and α and β are the two roots of the equation, then the factors of the equation can be expressed as

(x-α) and (x-β)

The same quadratic equation can be written in terms of these factors as

(x-α) (x-β) = 0

The Quadratic Formula

For a standard quadratic equation, ax2 + bx + c = 0, the quadratic formula is

x= -b b2-4ac2a

(b2 – 4ac), is called the discriminant of the equation and is represented by ‘D.’

• If D > 0, the quadratic equation will have unequal and real roots.

• If D = 0, the quadratic equation will have two equal and real roots.

• If D < 0, the quadratic equation will have complex roots.

Solved Examples

Example 1: Determine the roots of the quadratic equation x2+6x+3 = 0 using the quadratic formula.

Solution: x2+6x+3=0,

Finding the value of discriminant, D from the equation:

D= b2-4ac

D= 62-4.1.3=24

Because D>0, the equation will be real and distinct.

Using the quadratic formula:

x= -b b2-4ac2a

x= -6 62-4132

x= -6 242

x= -6 262

x= 2(-3 6)2

x= -36

Thus, the roots of the equation are -3+6 and -3-6.

Example 2: Use the factorisation method to solve the following quadratic equation

12x2+10x-12=0

Solution: The given quadratic equation is 12x2+10x-12 = 0.

Upon simplifying the equation, it becomes

6x2+5x-6=0

On splitting the middle term, we get

6x2 + 9x – 4x – 6 = 0

3x(2x+3) – 2(2x+3)=0

The factors of the equation are

(2x+3) and (3x-2)

Thus, the roots of the equation are

x=-3/2 and x= ⅔ 

Conclusion

The linear inequalities in a linear equation are those inequalities that cannot satisfy the whole equation. These inequalities are expressed with different inequality symbols such as <, >, ≥, and ≤. A quadratic equation is such an equation that can be expressed in the following form ax2+bx+c=0. The roots of a quadratic equation are those values of variable x that satisfy the equation. Roots can be found by a quadratic formula that is given by x= -b b2-4ac2a