- Quadratic Equation in One Variable is a mathematical sentence of degree 2 that can be written in the standard form .
- The members of the equation are quadratic term, linear term, and constant.
- The equation is a complete quadratic equation with .
- The equation is also a quadratic form but it is not simplified in standard form.
- In the case of , the quadratic equation is expressed in factored form. You need to use the FOIL method to write this in standard form.
- Finding the roots or solutions of a quadratic equation implies finding all permissible values of in .
- Quadratic equations in the form can be solved by applying the following concepts:
- If , then has two real solutions or roots, .
- If , then has one real solution or root, .
- If , then has no real solutions or roots.
- The principle in these concepts is facilitated using the Square Root Property: If , then .
Example 1. What are the roots of
Solving quadratic equations using Factoring:
- Apply the different ways of solving a quadratic trinomial of .
Example 1. Solve for the roots of .
Using factoring, we write the trinomial on the left side as .
Using the Zero Product Property, we have:
Example 2. Find the roots of .
In cases of , we use the ac test method.
To illustrate, we have:
Think of factors of whose sum is . The two factors are .
Using the factors of , we express the linear term of the equation as .
Take the common factors as .
Thus, the roots are
Solving quadratic equations using Po-Shen Loh Method:
Example 1. What are the roots of ?
For cases like , we divide each term of both sides of the equation by .
The two numbers whose sum is are .
The two roots of the equation is written as
Using the idea of product of , we have
Solving for yields . Use either
Example 1. Solve the equation using completing the square or quadratic formula.
Solution 1. Using Completing the Square
Rewrite the equation as .
Divide the coefficient of the linear term by 2 and square the result. The number obtained must be added to both sides of the equation.
Thus, the solutions are
Solution 2. Using the Quadratic Formula
Use the formula . By direct substitution of , we have:
The roots are
- Use the discriminant to determine the nature of the roots of a quadratic equation.
- If and not a perfect square, then there are two real, irrational roots.
- If and a perfect square, then there are two unequal real rational roots.
- If , then there is only one real and rational root.
- if , there are no real roots.
- To formulate a quadratic equation with the given roots , use the model .
- Completing the Square, Quadratic Formula, and Po-Shen Loh Method are applicable for quadratic equations having trinomials either factorable or non-factorable.
- Always be mindful of the signs when solving for the roots of the quadratic equation.
- When is written as , it is a common mistake to write the roots as . The roots must be .
- If you are asked to write the solution or roots of a quadratic equation, then use the term "or" instead of "and." For instance, we have: . It is incorrect to write it as .
- If you are asked to write the solution set of a quadratic equation, then write it as .
- Be careful in dealing with signs of the coefficients of when using the quadratic formula.