Quadratic Equations

Quadratic equations, also known as quadratics, are polynomial equations of the second degree. This implies that they consist of at least one term that is squared. A polynomial with degree two is a quadratic equation that can be expressed as:

ax² + bx + c = 0

in which x represents an unknown variable, and a, b, and c are constant values with a ≠ 0. The term ax² is the quadratic term, bx is the linear term, and c is the constant term.

1.0Quadratic Equation Definition

In algebra, the quadratic equation is defined as a polynomial equation of the second degree. This implies that they consist of at least one term that is squared. A quadratic equation with the variable x is an equation in the format:

ax² + bx + c = 0,

where a, b, and c denote real numbers, and a ≠ 0. For example, 2x² + x–300 = 0 is a quadratic equation. 

Similarly, 2x² – 3x + 1 = 0, 4x– 3 x² + 2=0 and 1–x² – 300 = 0 are also quadratic equations. Any equation is in the form p(x) = 0, where polynomial p(x) is of degree 2 and is a quadratic equation. 

Solutions to a quadratic equation can be determined by various methods, such as factoring, completing the square, or using the quadratic formula:    

$\mathrm{x}=\frac{-\mathrm{b}\pm \sqrt{{\mathrm{b}}^2-4\mathrm{ac}}}{2\mathrm{a}}$

Where ± denotes that there are typically two solutions unless the discriminant (b² – 4ac) is negative, in which case there are no real solutions. Quadratic equations often arise in various fields of mathematics, physics, engineering, and other sciences. They are also fundamental in algebra and serve as a basis for understanding more complex equations and functions.

2.0Standard Form of Quadratic Equations

The standard form of quadratic equation is:

ax² + bx + c = 0

In this form:

• x represents the variable.
• a, b, and c denote constant, with a not equal to zero.

The standard form helps identify the coefficients a, b, and c and for solving quadratic equations using various methods like factoring, completing the square, or using the quadratic formula.

3.0Quadratic Equation Formula

The quadratic formula is a fundamental tool for solving quadratic equations. It provides the roots (or solutions) of a quadratic equation of the form:

ax² + bx + c = 0

The quadratic formula is:

$\mathrm{x}=\frac{-\mathrm{b}\pm \sqrt{{\mathrm{b}}^2-4\mathrm{ac}}}{2\mathrm{a}}$

• x represents the variable (the unknown).
• a, b, and c denote constants with a ≠ 0.
• b² – 4ac is called the discriminant.

The discriminant determines the nature of the roots:

• If b² – 4ac > 0, then there exist two distinct real roots.
• If b² – 4ac = 0, then there is one distinct real root (the equation exhibits repeated roots).
• If b²– 4ac < 0, then there is no real root (the roots are complex).

The " addition and subtraction " (±) sign in the formula indicates that there are typically two solutions for x, one obtained by adding the square root term and the other by subtracting it.

The quadratic formula is used as a powerful tool for solving quadratic equations quickly and accurately, even when factoring or completing the square methods is not convenient.

4.0Quadratic Equation Examples

Here are some examples of quadratic equations.

• x²+ 2x + 1 = 0
• 2x² + x + 1 = 0
• x² + 3x + 1 = 0
• –x² + 3x + 5 = 0
• 7x² + x + 2 = 0

5.0How To Solve Quadratic Equations

The following methods can be used to solve quadratic equations.

1. Factorization
2. Completing the square
3. Using the Quadratic Formula

1. Factorization

If the quadratic expression is factorable, you can factor it into two binomial expressions and set each factor equal to zero to find the solutions.

Example: Solve:    x² – 5x + 6 = 0

Factorize as (x – 2) (x – 3) = 0

So, the solutions are x = 2 and x = 3.

2. Completing the square

You can rewrite the quadratic equation in the form (x + p)² = q, where p and q are constants, and then solve for x. 

Example: Solve: x² – 4x – 5 = 0 

First, move the constant term to the other side:  x² – 4x = 5 

Next, add (4/2)² = 4 to both sides to complete the square:

x² – 4x + 4 = 5 + 4

This simplifies to (x – 2)² = 9. 

Taking the square root of both sides, we get x – 2 = ± 3, so the solutions are x = –1 and x = 5 

3. Using the Quadratic Formula

The quadratic formula is a general method that can be applied to any quadratic equation. It states that for the equation ax²+bx+c=0, the solutions are given by:

$\mathrm{x}=\frac{-\mathrm{b}\pm \sqrt{{\mathrm{b}}^2-4\mathrm{ac}}}{2\mathrm{a}}$

Example: Solve: 2x² + 3x – 2= 0

we have a = 2, b= 3, and c = –2.  Substituting these values into the quadratic formula:

$x=\frac{-3\pm \sqrt{3^2-4\cdot 2\cdot (-2)}}{2\cdot 2}$

$x=\frac{-3\pm \sqrt{9+16}}{4}$

$x=\frac{-3\pm \sqrt{25}}{4}$

$x=\frac{-3\pm 5}{4}$

So, the solutions are

$\frac{-3+5}{4}=\frac{1}{2}\text{ and }\frac{-3-5}{4}=-2$

6.0Relation Between Roots and Coefficient

1. If α and β are the roots of the quadratic equation, 

ax² + bx + c = 0

sum of the roots, α + β = –b/a = – coefficient of x /coefficient of x²

and product of the roots,  αβ = c/a = constant term /coefficient of x² 

Find Quadratic Equation from roots

If we want to find the quadratic equation from roots, then

Let α and β be two roots then the quadratic equation will be

x² – (α + β)x + αβ = 0

x² - (sum of roots )x + (product of the roots) = 0

7.0Quadratic Equation JEE Questions

Question 1. If  α ≠ β but α² = 5α – 3 and  β² = 5β – 3 then the equation having α/β and β/α as its roots is

a. 3x² – 19x + 3 = 0 c. 3x² +19x –3 =0

b. x² – 19x – 3 = 0 d. x² – 5x +3 = 0

Ans: - We have α² = 5α – 3 and β² = 5β – 3; ⇒ α and β are roots of equation, x² = 5x –3 or x² = 5x – 3 or x² – 5x + 3 = 0

∴  α + β = 5 and αβ = 3

Thus, the equation having α/β and β/α as its roots is x² – (α/β + β/α)x + αβ/αβ = 0

x² – {(α2 + β2)/αβ}x + 1 = 0

3x² – 19x +3 = 0

Question 2. The quadratic equations x² – 6x + a = 0, x² – cx + 6 = 0 have 1 root in common. The other roots of the first and second quadratic equations are integers in the ratio 4 : 3. Then, the common root is

a. 2 b. 1 c. 4 d. 3

Ans: Let the common root be α

As the other roots are in 4 : 3

so, Let 4β is the root of x² – 6x + a = 0 and 3β is the root of x² – cx + 6 = 0

So, by this α + 4β = 9 and 4αβ = a

 α + 3β = c and 3αβ = 6

as, 3αβ = 6

4 ✕  2 = a ⇒ 8 = a

The first equation is x² – 6x + 8 = 0

We can rewrite it as  x² – 2x – 4x + 8 = 0

x(x –2) – 4(x –2) = 0

(x –2)(x –4) = 0

So, x – 2 = 0 and x –4 = 0

x = 2 and x = 4

So, The value of x is 2, 4.

For α = 2, 4β = 4

So, β = 4/4, β  = 1

3β = 3

Hence the common root is α = 2.

8.0Solved Questions for Quadratic Equation

1. What is a quadratic equation?

Ans: The quadratic equation is defined as a polynomial equation of the second degree, written in the format ax² + bx + c = 0, where x represents an unknown variable and a, b, and c denote constants with a ≠ 0.

2. What is the quadratic formula?

Ans: The quadratic formula is a general formula used to find the solutions of any quadratic equation. It states that for the equation ax² + bx + c = 0, the solutions are given by:   

$\mathrm{x}=\frac{-\mathrm{b}\pm \sqrt{{\mathrm{b}}^2-4\mathrm{ac}}}{2\mathrm{a}}$