What are 5 methods of Solving a Quadratic Equation ?
A quadratic equation is a polynomial equation in a single variable of the form where
A quadratic equation has two solutions. Either two distinct real solutions, one double real solution or two imaginary solutions. There are several methods you can use to solve a quadratic equation.
5 methods of Solving a Quadratic Equation
(i) Factoring(split the middle term)
(ii)Principle of square roots
(iii)Completing the Square
(iv) Quadratic Formula
(v) Graphing
Contents
Factoring(split the middle term):
Factoring a quadratic equation by splitting the middle term involves breaking the middle term of the quadratic expression into two terms, which then allows you to factor by grouping. The steps are as follows:
1. Start with a quadratic equation in the form where
2. Identify
3. Split the middle term (
4. Group the terms in pairs:
5. Factor out the common factor from each group:
6. Factor out the common binomial factor:
7. Set each factor equal to zero and solve for
So, the solutions to the quadratic equation
Principle of square roots :
The square roots method is one of the methods used to solve quadratic equations. The general form of a quadratic equation is:
The square roots method is applicable when the quadratic equation can be expressed in the form:
To solve for
This gives you two possible solutions for the quadratic equation. The
It’s important to note that not all quadratic equations can be easily factored into the form
Example :
Solution: Moving the constant to the right side , we get
Take the square root of both sides
So the solution of given equation is
Example :
Solution :
Completing the Square method of solving quadratic equation:
Here’s a step-by-step guide on how to use the completing the square method:
1. Start with the quadratic equation .
2. If to make the coefficient of
3. Rearrange the equation to isolate the terms on one side and move the constant term to the other side.
4. Add and subtract
6. Take the square root of both sides and solve for
This is the solution to the quadratic equation using the completing the square method. Note that sometimes the square root term simplifies to a real number, while other times it results in complex numbers.
Example : solve the quadratic equation using the completing the square method.
Solution : Given that
Since a=2, divide the entire equation by 2. We get
Rearrange the equation to isolate the
So, the solutions to the quadratic equation
Quadratic Formula for solving a quadratic equation
Consider a quadratic equation in the standard form
The quadratic formula gives the solutions for
The expression under the square root, is called the discriminant (
- If
, the quadratic equation has two distinct real roots.
- If
, the quadratic equation has one real root (the parabola touches the x-axis at one point).
- If
the quadratic equation has two complex roots (conjugate pairs).
Example : Find the roots of the equation .
Here
Now, let’s apply the quadratic formula:
So the solutions are
Therefore the solutions are
Graphing method for solving a quadratic equation
For any quadratic polynomial , the graph of the corresponding equation
The solutions of the equation are the 𝑥 values for which the function is zero, which we refer to as the roots of the function. On a graph, these values are the 𝑥-coordinates of the points where the 𝑦-value is zero, which corresponds to the points at which the graph crosses the 𝑥-axis.
A quadratic equation will have up to two real solutions. If an equation has two solutions, the corresponding function will have a graph that crosses the 𝑥-axis twice. An equation with a repeated solution will lead to a graph that has a vertex on the 𝑥-axis. Finally, an equation having no solution will mean that the graph is entirely above or below the 𝑥-axis.
In order to find the solutions of a quadratic equation using a graph:
(i) Rearrange the equation so that one side (if necessary).
(ii) Draw the graph of the quadratic function.
(iii) Read off the -coordinate(s) of the point(s) where the curve crosses the -axis.
Example : Find the solutions of the equation graphically.
Solution: Draw the graph of
![](https://www.mathmitra.com/wp-content/uploads/2024/05/Untitled-1-300x271.webp)
Now read off the -coordinate(s) of the point(s) where the curve crosses the -axis.
Here we have one (repeated) root at
Related post:
- Zeros of quadratic Polynomials
- Factoring trinomials worksheet
- Relationship between Zeros and coefficients of a Polynomial