The standard form for a quadratic equation is ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0; for example: x2 + 6x + 5 = 0, 3x2 - 2x = 0, and x2
Some equations can be solved by factoring. To do this, first add or subtract expressions to bring all the expressions to one side of the equation, with 0 on the other side. Then try to factor the nonzero side into a product of expressions. Each of the factors can be set equal to 0, yielding several simpler equations that possibly can be solved. The solutions of the simpler equations will be solutions of the factored equation.
As an example, consider the equation: x2 – 7x = - 12
x2 – 7x + 12 = 0 (taking all terms on one side and putting the expression equal to zero)
Now try to break b into two parts, such that the sum of those two parts = ‘b’ and the product is equal to the product of ‘a’ and ‘c’.
x2 – 4x – 3x + 12 = 0
⇒ x (x – 4) – 3(x – 4) = 0
⇒ (x – 3) (x – 4) = 0.
Putting these separately equal to 0
⇒ x – 3 = 0, x = 3 and x – 4 = 0, x = 4.
Thus the solutions of the equation are 3 and 4.
The solutions of an equation are also called the roots of the equation.
A quadratic equation has at most two real roots and may have just one or even no real root. For example, the equation x2 – 6x + 9 = 0 can be expressed as (x – 3)2 = 0, or (x – 3) (x – 3) = 0; thus the only root is 3. The equation x2 + 4 = 0 has no real root; since the square of any real number is greater than or equal to zero, x2 + 4 must be greater than zero.
An expression of the form a2 – b2 can be factored as
(a – b)(a + b)
For example, the quadratic equation 9x2 – 25 = 0 can be solved as follows
(3x – 5)(3x + 5) = 0
3x – 5 = 0 or 3x + 5 = 0
x = 5/3 or x = -5/3
Must Read Quadratic Equations Articles
Solving Equations by using the Quadratic Formula:
If a quadratic expression is not easily factored, then its roots can always be found using the quadratic formula: If ax2 + bx + c = 0 (a ≠ 0), then the roots are
These are two distinct real numbers unless b2 – 4ac < 0.
If b2 – 4ac = 0; then these two expressions for x are equal to –b/2 a and the equation has only one root.
If (b2 – 4ac) < 0, then √b2 – 4ac is not a real number and the equation has no real roots.
To solve the quadratic equation x2 – 7x + 8 = 0 using the above formula, note that a = 1, b = - 7, and c = 8, and hence the roots are
b2 – 4ac is called the discriminant and is denoted by the symbol Δ or is represented by the letter D. Following are some of the important points relating to the discriminant and its relation with the nature of the roots.
- If Δ > 0, then both the roots will be real and unequal and the value of roots will be If Δ is a perfect square, then roots are rational otherwise they are irrational.
- If Δ = 0, then roots are real, equal and rational. In this case the value of roots will be –b/2a.
- If Δ < 0, then roots will be imaginary, unequal and conjugates of each other.
- If α and β are the roots of the equation ax2+ bx + c = 0, then sum of the roots i.e. α + β = -b/a
- If α and βare the roots of the equation ax2+ bx + c = 0, then product of the roots i.e. αβ = c/a
- Ifα and β, the two roots of a quadratic equation are given, then the equation will be x2– (α + β)x + αβ = 0.
The equation is x2– (sum of roots)x + product of roots = 0
These were some very important points relating to the quadratic equations. The following are some properties regarding the roots of the equation.
- If in the equation b = 0, then roots are equal in magnitude, but opposite in sign.
- If a = c, then roots are reciprocal of each other.
- If c = 0, then one of the roots will be zero.
- If one root of a quadratic equation be a complex number, the other root must be its conjugate complex number i.e.
α = j + √-k then β = j-√-k ⇒ α = j + ik and β = j - ik