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In mathematics, a quadratic problem is one in which a variable is multiplied by itself, which is known as squaring.
In this language, the size of a square is determined by multiplying its side length by itself. The term “quadratic” is derived from the Latin quadratum, which meaning “square.”
Quadratic equations describe a wide range of real-world occurrences, including where a rocket ship will land, how much to charge for a product, and how long it takes a person to paddle up and down a river. Quadratics have a long history and are fundamental to the history of algebra because of their vast range of applications.
What is a Quadratic Equation?
A quadratic equation is a second-degree algebraic equation in x. The conventional form of the quadratic equation is ax2 + bx + c = 0, with a and b as coefficients, x as the variable, and c as the constant component. The coefficient of x2 is a non-zero term(a ≠0), which is the first requirement for determining whether or not an equation is quadratic. When writing a quadratic equation in standard form, the x2 term comes first, followed by the x term, and finally the constant term. Integral values, rather than fractions or decimals, are commonly used to express the numeric values of a, b, and c.
Formula for a Quadratic Equation
The most easy way for calculating the roots of a quadratic equation is to use the Quadratic Formula. Some quadratic equations are difficult to factor, and we can use this quadratic formula to acquire the roots as rapidly as possible in these circumstances. The sum and product of the roots of a quadratic equation can also be found using the roots. The two roots of the quadratic formula are given as a single equation in the quadratic formula. Using either the positive or negative sign, the equation’s two unique roots can be found.
x = [-b ± √(b² – 4ac)]/2a.gives the roots of a quadratic equation ax2 + bx + c = 0.
Applications Of The Quadratic Equations
Quadratic equations are used to solve many physical and mathematical problems. The solution of the quadratic equation is particularly important in mathematics. As previously established, a quadratic equation with D 0 has no valid solutions. As you will see in the following sessions, this scenario is critical. It contributes to the development of Complex Analysis, a separate discipline of mathematics.
Quadratic equations can be found in a variety of fields. By examining a few cases, we will attempt to describe a few uses. Let’s get this party started.
Application to Problems of Area
Example 1: There is a hall that is five times longer than it is wide. The floor is 45m2 in size. Determine the hall’s length and width.
Solution: Assume that ‘w’ represents the hall’s width. Then we observe that w (5w) equals the hall’s area. As a result, we can write:
5w2 = 45
w2 = 9
w2 – 9 = 0
(w+3)(w-3) = 0
w = -3 or w = 3 is a mathematical expression. As a result, the width is 3 metres and the length is 5(3) metres.
Example 2: A right-angled triangle has three sides: x, x+1, and 5. If the longest side is 5, find x and the area.
The Hypotenuse will be the longest side of the triangle. As a result, we can write:
x2 + (x+1)2 = 52 (Pythagoras’ Theorem)
x2 + x2 + 2x + 1 = 25
2x2 + 2x – 24 = 0
As a result, x2 + x – 12 = 0.
(x – 3)(x + 4) = 0
(x + 4) = 0 or (x – 3) = 0 is the answer.
x = -4 or x = 3 are two possibilities.
Because the length cannot be negative, we can only choose x = 3. (Why?)
As a result, x = 3 and Area = 1/2 x 3 x 4 = 6
Tips to Solve Equations reducible to Quadratic
• To solve the equations of type ax4 + bx2 + c = 0, put x2 = y
• To solve a.p(x)2 + b.p(x) + c = 0, put p(x) = y.
• To solve a.p(x) + b/p(x) + c = 0, put p(x) = y.
• To solve a(x2 + 1/x2) + b(x + 1/x) + c = 0,put x + 1/x = y and to solve a(x2 + 1/x2) + b(x – 1/x) + c = 0, put x – 1/x = y.
• To solve a reciprocal equation of the type ax4 + bx3 + cx2 + bx + a = 0, a ≠ 0, divide the equation by d2y/dx2 to obtain a(x2 + 1/x2) + b(x + 1/x) + c = 0,and then put x + 1/x = y.
• To solve (x + a) (x + b) (x + c) (x + d) + k = 0 where a + b = c + d, put x2 +(a + b)x = y
To solve an equation of type √(ax + b) = cx + d or √(ax2 + bx + c) = dx + e, square both the sides.
• To solve √(ax + b) ± √(cx + d) = e, transfer one of the radicals to the other side and square both sides. On one side, keep the expression with the radical sign, and on the other side, transfer the remaining expression.
Conclusion:-
In everyday life, we employ quadratic formulas to calculate areas, determine the profit of a product, and calculate the speed of an object. Furthermore, a quadratic equation is one that involves at least one squared variable.
