- 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.
Solved Examples (Extraction of Square Roots)
- 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
Solved Examples (Factoring and the Po-Shen Loh Method)
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
Solved Examples (Completing the Square and Quadratic Formula)
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.
- Keith Madrilejos
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