SafekipediaIn mathematics, a quadratic equation is a special kind of math problem that helps us find unknown numbers. It looks like this: a × x² + b × x + c = 0, where x is the unknown number, and a, b, and c are numbers we already know. The letter a must not be zero, or the equation would not be quadratic.
Quadratic equations are useful because they can have up to two answers, called solutions. These solutions are also named roots or zeros. Sometimes, both answers are the same, and this is called a double root.
We can solve quadratic equations in different ways. One way is by using the quadratic formula: x = (−b ± √(b² − 4a**c)) / (2a). This formula gives us the answers directly. People have been solving problems that lead to quadratic equations for thousands of years, as early as 2000 BC.
Solving the quadratic equation
A quadratic equation can have zero, one, or two solutions, also called roots. When there is only one distinct root, it is called a double root. When there are no real roots, the equation still has two complex-valued roots.
There are several ways to solve a quadratic equation. Factoring by inspection is often the first method taught. It means rewriting the equation as a product of two simpler expressions and solving each one. Completing the square is another method that changes the equation into a perfect square, making it easier to solve. Finally, the quadratic formula gives a direct way to find the solutions by using the coefficients of the equation.
The discriminant, a part of the quadratic formula, helps tell us about the roots. A positive discriminant means two real roots, zero means one real root (a double root), and a negative discriminant means two complex roots.
Examples and applications
Quadratic equations are used in many areas. For example, they help find the golden ratio, which appears in art and architecture. They also describe shapes like circles, ellipses, parabolas, and hyperbolas.
In physics, quadratic equations help predict the movement of objects with constant acceleration. They are also important in chemistry for calculating the pH of certain solutions.
History
Babylonian mathematicians as early as 2000 BC could solve problems about the areas and sides of rectangles. They used a method to find the answers to equations that look like quadratic equations today.
Later, mathematicians in Egypt, Greece, China, and India also used geometric methods to solve quadratic equations. In the 7th century AD, an Indian mathematician named Brahmagupta gave one of the first known formulas for solving these equations. In the 9th century, Muhammad ibn Musa al-Khwarizmi developed formulas that worked for positive answers and described a method called "completing the square." Over time, mathematicians began to accept negative and irrational numbers as answers to quadratic equations. By the 17th century, the quadratic formula was written in the form we use today.
Advanced topics
Alternative methods of root calculation
Vieta's formulas
Main article: Vieta's formulas
Vieta's formulas connect the roots of a quadratic equation to its numbers. They show that the total of the roots is the opposite of the middle number divided by the first number. The multiply of the roots is the last number divided by the first number. These formulas help us understand patterns in quadratic graphs.
Trigonometric solution
Before calculators, people used math tables to solve problems. Astronomers needed fast ways for hard math. One old way used angles to solve quadratic equations. This could save time compared to using tables alone. This way turned the problem into a shape with angles and then used tables to find the answer.
Solution for complex roots in polar coordinates
Quadratic equations with real numbers can sometimes have unusual answers. There are picture ways to solve these using shapes like circles. One way, called Lill's method, uses a four-sided shape and a circle to find the answers. Another way, the Carlyle circle, uses a circle to find the answers by seeing where the circle meets the horizontal axis.
Generalization of quadratic equation
The usual ways to solve quadratic equations work even if the numbers are unusual or from special number groups. In some special number groups where the number 2 acts in a different way, the standard way doesn’t work, but there are other ways to find the answers using different math ideas.
| a x 2 + b x ± c = 0 , {\displaystyle ax^{2}+bx\pm c=0,} | 1 |
| x = c / a tan θ {\displaystyle x={\textstyle {\sqrt {c/a}}}\tan \theta } | 2 |
| tan 2 θ n = + 2 a c b , {\displaystyle \tan 2\theta _{n}=+2{\frac {\sqrt {ac}}{b}},} | 4 |
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