A quadratic equation has two solutions that may be or may not be distinct. The result may be real numbers or imaginary numbers. Learn the important formulas of quadratic equation, definition here. Let us learn about the introduction to the quadratic equations from this article. You can find examples of quadratic equations with step-by-step explanations.
What is a Quadratic Equation?
In the name, quadratic “quad” means square because the equation is square. A quadratic equation is an algebraic expression of the 2nd degree in variable x. The variable x has two answers real or complex numbers. The answers or solutions of x are called roots of the quadratic equations. They are specified as (α, β). The standard form of the quadratic equation is ax² + bx + c = 0. Where a, b is the coefficient of x² and c is the constant. a,b, c are not fractions nor decimals.
Quadratic Equation Formula
The formula for the quadratic equation is an easy method to find the roots of the equation. Without the formulas, the values are not factorized, and can find the roots in the easiest way. The roots of Q.E helps to find the sum of the roots and product of the roots of the quadratic equation.
Quadratic Equation (α, β) = [-b ± √(b² – 4ac)]/2a.
Important Formulas to Solve Quadratic Equations
- The standard form of the quadratic equation is ax² + bx + c = 0.
- The discriminant(D) of quadratic equation is D = b² – 4ac.
- For the case, D = 0 the roots are real and equal.
- For the case, D > 0 the roots are real and distinct.
- For the case, D < 0 the roots do not exist, or the roots are complex.
- The product of the Root of the quadratic equation is αβ = c/a = Constant term/ Coefficient of x²
- The roots of the quadratic equation is x = [-b ± √(b² – 4ac)]/2a.
- The sum of the roots of a Q.E is α + β = -b/a = – Coefficient of x/ Coefficient of x²
- Quadratic equation in the form of roots is x² – (α + β)x + (αβ) = 0
- If α, β, γ are roots of a cubic equation ax³ + bx² + cx + d = 0, then, α + β + γ = -b/a, αβ + βγ + λα = c/a, and αβγ = -d/a
- The roots (α + iβ), (α – iβ) are the conjugate pair of each other.
- For a > 0, the quadratic expression f(x) = ax² + bx + c has a minimum value at x = -b/2a
- For a < 0, the quadratic expression f(x) = ax² + bx + c has a maximum value at x = -b/2a
- For a > 0, the range of the quadratic equation ax² + bx + c = 0 is [b² – 4ac/4a, ∞).
- For a < 0, the range of the quadratic equation ax² + bx + c = 0 is (∞, -(b² – 4ac)/4a]
Methods for Solving Quadratic Equations
There are three methods for solving quadratic equations. They are as follows,
1. Factorization method
2. Completing the square method
3. Quadratic Equation formula
Quadratic Equation Question and Answers
Example 1.
Solve 5x² + 7x + 2 = 0
Solution:
Coefficients are: a = 5, b = 7, c = 2
x = [-b ± √(b² – 4ac)]/2a
x = [-7 ± √(7² – 4.5.2)]/2.5
x = [-7 ± √(49 – 40)]/10
x = [-7 ± √(9)]/10
x = [-7+3]/10 = -4/10 = -2/5
x = [-7 – 3]/10 = -10/10 = -1
Thus x = -2/5 or x = -1
Example 2.
Find the range of k for which 4 lies between the roots of the quadratic equation x² + 2(k – 4)x + 5 = 0.
Solution:
6 will lie between the roots of the quadratic expression f(x) = x² + 2(k – 4)x + 5 if,
f(4) < 0
= 16 + 2(k – 4)4 + 5 < 0
= 16 + (2k – 8)4 + 5 < 0
= 16 + 8k – 32 + 5 < 0
= 8k – 11 < 0
= k < 11/8
Example 3.
Find the factors of the quadratic equation x² + 7x + 12 = 0
Solution:
x² + 7x + 12 = 0
x² + 3x + 4x + 12 = 0
x(x + 3) + 4(x + 3) = 0
(x + 4) (x + 3) = 0
x + 4 = 0 or x + 3 = 0
x = -4 or x = -3
FAQs on Quadratic Equation
1. What is the purpose of quadratic equations?
Quadratic equations are actually used in our daily life, as when calculating areas, determining a product’s profit or formulating the speed of an object.
2. What is the standard form of the quadratic equation?
The standard form of the quadratic equation is ax² + bx + c = 0
3. How many roots does a quadratic equation have?
The quadratic equation has two roots. The Q.E with real or complex coefficients has two solutions that are called roots.