The word quadratic is derived from the Latin word “quadratum” which means “A Square Figure”. A polynomial of the form ax² + bx + c, where a ≠ 0 is a quadratic polynomial or expression in the variable x of degree 2. If a quadratic expression ax² + bx + c is equated to zero, it become a quadratic equation. Simply we can say, a quadratic equation in the variable x is an equation of the form ax² + bx + c = 0, where a, b, c, are real numbers and a ≠ 0. Equation ax² + bx + c = 0, a ≠ 0 is called the standard form of a quadratic equation whereas y = ax² + bx + c is called a quadratic function. If the quadratic equation has variable in second degree only, it is called pure quadratic equation. If it has variable with both second degree and first degree, it is called adfected quadratic equation.
How do we solve a Quadratic Equations?
Solving a quadratic equation means, finding the roots of quadratic equation. The roots can be verified by substituting the values in the quadratic equation and checking whether they satisfy the equation. The roots also from the solution set of the quadratic equation.
What is Factorisation Method of Quadratic Equation?
Factorisation Method is used when the quadratic equation can be factorise into two linear factors. After the Factorisation, the quadratic equation is expressed as the product of its two linear factors and this is equated to zero to get the solutions.
Steps of finding roots of quadratic equations by Factorisation Method:
- Write the given equation in standard form of quadratic equation, ax² + bx + c = 0.
- Resolve the quadratic equation expression (LHS) by splitting its middle term.
- Take the common factor and obtain the two linear factors.
- Equate each factor to zero.
- Simplify each linear equation and find the value of unknown.
How do we use the Completing the Square Method for Quadratic Equations?
The method of completing the square is one the best method of find the roots of a quadratic equation. The idea behind this method is to adjust the left side of the quadratic equation so that it becomes a perfect square. Steps of the quadratic equation by ax² + bx + c = 0 by completing the square method:
- Divide each side by “a”.
- Rearrange the equation so that constant term c/a is on the right side (RHS).
- Add [½ (b/a)]² to both sides to make LHS, a perfect square.
- Write the LHS as a square and simplify the RHS.
- Solve it.
How do we use Quadratic Formula?
Steps for solving a quadratic equation using the quadratic formula:
- Write the equation in standard form ax² + bx + c = 0.
- Compare the equation with standard form and identify the values of a, b and c.
- Write the quadratic formula x = [-b ± √(b² – 4ac)]/2a.
- Substitute the values of a, b and c in the formula.
- Simplify and get the two roots.
Uses of Quadratic Equation or Quadratic Functions.
Some of the uses of quadratic equations are as follow:
- When the rocket is fired upward, then the height of the rocket is defined by a “quadratic function”.
- Shapes of the satellite dish, reflecting mirror in a telescope, lens of the eye glasses and orbits of the celestial objects are defined by the quadratic equations.
- The path of a projectile is defined by quadratic function.
- When the breaks are applied to a vehicle, the stopping distance is calculated by using quadratic equation.
Conditions for the roots of a Quadratic Equation
A quadratic equation ax² + bx + c = 0 has
- two distinct real roots, if b² – 4ac > 0,
- two equal roots (i.e., coincident roots), if b² – 4ac = 0, and
- no real roots, if b² – 4ac < 0.
History of Quadratic Equations
Solving of a quadratic equation is often credited to ancient Indian Mathematicians Brahmagupta and Sridharacharya. It is believed that the solutions of a quadratic equations in general form was introduced by an Indian Mathematician Brahmagupta. He gave an explicit formula to solve a quadratic equation of the form ax² + bx = c. In A.D. 1025 Sridharacharya derived a formula (known as quadratic formula) for solving a quadratic equation by the method of completing the square. It is believed that Babylonians were the first to solve quadratic equations. Greek Mathematician Euclid develop a geometrical approach for finding length, which are nothing but solutions of quadratic equations. An Arab mathematician Al-khwarizni also studied quadratic equations of different types.
What is the meaning of a quadratic equation?
The word quadratic is derived from the Latin word “quadratum” which means “A Square Figure”. If a quadratic expression ax² + bx + c is equated to zero, it become a quadratic equation.
How do you solve quadratic equations?
There are various methods to solve a quadratic equation:
1. Factorization Method
2. Completing the square Method
3. Quadratic Formula Method
What is the quadratic equation used for?
The Quadratic equation is used for day to day life use. Most of the application based on distance-time, work-time, Cost of items and number of items, etc. are based on quadratic equation.
What are the 4 ways to solve quadratic equations?
The 4 ways to solve quadratic equations are as follows:
1. Completing the Square.
2. Factorization Method
3. Remainder theorem method
4. Using Quadratic Formula