Basics of Quadratic Equations

We can define Quadratics as a polynomial equation of a second degree, which shows that it contains a minimum of one term that is squared. The word “quadratic” comes from the word quad which means square because any one of the variables is squared like a2, x2, b2, etc.

It is also called quadratic equations or equations of degree 2.

Standard Form of a Quadratic Equation

The standard or general form of a quadratic equation is: –

ax² + bx + c = 0

Here, x is an unknown variable and a, b, and c are the known numeric values.

Note – a cannot be equal to 0, as if a is o it will not remain a quadratic equation. It will become a linear equation ( bx + c ).

For eg :    –  3x² + 7x +9

Here,   a = 3
b = 7
c = 9

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Roots of Quadratic Equation

Roots are the values of variables (x,y, etc) that satisfies a given quadratic equation. In simple words, y = α is a root of the quadratic equation f(y), if f(α) = 0.

The real roots of equation f(x) = 0 are the x-coordinates of the points where the curve y = f(x) intersects the x-axis.

Properties of Roots

• One root of the quadratic equation is zero and the other root is -b/a if c = 0
• If b = c = 0, both the roots will be zero.
• If a = c, the roots will be reciprocal to each other.

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Quadratuce Equation Formula

The Roots of a quadratic equation are given by a formula:

(α, β) = [-b ± √(b² – 4ac)]/2a

This formula is known as the quadratic formula.

According to the general form: –

ax² + bx + c = 0

• a , b and c are constants, α, β are the roots of a quadratic equation.
• D = discriminant, which is equal to: –
  D = b² – 4ac

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What is a Discriminant?

The term b² – 4ac in the quadratic equation formula is known as the discriminant of a quadratic equation. It describes the nature of roots.

Let’s study the Nature of the Roots of Quadratic Equation:

1. If the value of the Discriminant is 0, the quadratic equation will have equal roots which means if b² – 4ac = 0
   0 , α = β.
2. When the value of the Discriminant < 0, it will have imaginary roots.
3. When the value of the Discriminant > 0, it will have real roots.
4. When the value of the Discriminant > 0, and is a perfect square, the quadratic equation will have rational roots.
5. When the value of the Discriminant > 0, and is not a perfect square, the quadratic equation will have irrational roots.

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As we can see, there is a relation between the roots and coefficients of the quadratic equation.

The Relationship Between Roots and Coefficients are as Follows.

In a Quadratic Equation ax² + bx + c = 0, the roots are α and β then,

1. If α + β, the roots will be -b/a.
2. If αβ, the roots will be c/a.
3. If α – β, the roots will  be±√[(α + β)² – 4αβ].
4. If |α + β|, it will be equal to √D/|a|.

In the next session, we will discuss how to solve Quadratic Equation by different methods.