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What is Quadratic Equation?
A quadratic equation is a second-order polynomial equation in a single variable x with a ≠ 0. It can be written in the form:
ax2+bx+c= 0
Where x represents an unknown, and a, b, and c represent known numbers, with a ≠ 0. If a were 0, the equation would be linear, not quadratic.
The solutions to the quadratic equation are known as the roots of the equation. They can be calculated using various methods, including:
- Factoring: Expressing the quadratic in a product of two binomials, if possible, and using the zero product property to find the roots.
- Completing the square: Rewriting the equation in the form of a perfect square trinomial, which can then be solved for x.
- Quadratic formula: The most general method, which can solve any quadratic equation. The roots are given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where the ± symbol indicates that the quadratic equation has two solutions, and \( \sqrt{b^2 - 4ac} \) is called the discriminant. The discriminant can determine the nature of the roots:
- - If \( b^2 - 4ac > 0 \), there are two distinct real roots.
- - If \( b^2 - 4ac = 0 \), there is exactly one real root (also called a double root).
- - If \( b^2 - 4ac
Quadratic equations are fundamental to algebra and appear in various applications across mathematics, physics, engineering, and economics.
How to solve quadratic equations?
Solving quadratic equations can be approached using several different methods, depending on the form of the quadratic equation and the specific values of the coefficients \( a \), \( b \), and \( c \). Here are the most common methods:
1. Factoring:
If the quadratic equation can be factored into the product of two binomials, it is often the simplest method.
For example, to solve \( x^2 - 5x + 6 = 0 \):
- Factor the quadratic into \((x - 2)(x - 3) = 0\).
- Apply the zero product property, which states that if a product of factors equals zero, at least one of the factors must be zero. So, set each factor equal to zero: \( x - 2 = 0 \) or \( x - 3 = 0 \).
- Solve for \( x \): \( x = 2 \) or \( x = 3 \).
2. Completing the Square:
This method involves rewriting the quadratic in the form of a perfect square trinomial.
- Begin with \( ax^2 + bx + c = 0 \).
- Divide through by \( a \) to get \( x^2 + \frac{b}{a}x = -\frac{c}{a} \).
- Add \( \left(\frac{b}{2a}\right)^2 \) to both sides to complete the square on the left side.
- Write the left side as a square and simplify the right side.
- Take the square root of both sides and solve for \( x \).
3. Quadratic Formula:
This formula can solve any quadratic equation, regardless of the coefficients. Given \( ax^2 + bx + c = 0 \), the solutions for \( x \) are: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here’s how you apply it:
- Identify the coefficients \( a \), \( b \), and \( c \) in the quadratic equation.
- Plug them into the quadratic formula.
- Simplify under the square root, known as the discriminant \( \Delta = b^2 - 4ac \).
- Evaluate the two possible solutions using the \( + \) and \( - \) signs.
4. Graphical Method
You can also solve quadratic equations by graphing the quadratic function \( y = ax^2 + bx + c \) and finding the points where the graph intersects the x-axis. The x-coordinates of these points are the solutions to the quadratic equation.
Each method has its advantages in different situations, and the choice of method can be influenced by the simplicity of the arithmetic, the form of the quadratic equation, and the solver's personal preference.
Application of quadratic equations?
Quadratic equations are a fundamental tool in various fields of science, engineering, economics, and many other areas.
See also