Algebra 1 - Quadratic Equations


  • Quadratic Equation in One Variable is a mathematical sentence of degree 2 that can be written in the standard form ax2+bx+c=0, where a,b,c and a0.
  • The members of the equation are ax2=quadratic term, bx=linear term, and c=constant.
  • The equation 3x2+5x-2=0 is a complete quadratic equation with a=3, b=5, c=-2.
  • The equation 2xx-9=0 is also a quadratic form but it is not simplified in standard form.
  • In the case of 4x-32x+1=0, 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 x in ax2+bx+c=0.
  • Quadratic equations in the form x2=b can be solved by applying the following concepts: 
  • If b>0, then x2=b has two real solutions or roots, x=±b.
  • If b=0, then x2=b has one real solution or root, x=0.
  • If b<0, then x2=b has no real solutions or roots.
  • The principle in these concepts is facilitated using the Square Root Property: If x2=b and b0, then x=±b.


Example 1. What are the roots of 3x-52=135?






x=35+5 or x=-35+5

Solved Examples (Factoring and the Po-Shen Loh Method)

Solving quadratic equations using Factoring:

  • Apply the different ways of solving a quadratic trinomial of ax2+bx+c=0 where a=1 or ax2+bx+c=0 where a1


Example 1. Solve for the roots of a2-11a-26=0.


Using factoring, we write the trinomial on the left side as a-13a+2=0.

Using the Zero Product Property, we have:


a=-2 or a=13


Example 2. Find the roots of 3x2+11x-20=0.


In cases of ax2+bx+c=0 where a1, we use the ac test method.

To illustrate, we have:


a=3, b=11, c=-20


Think of factors of -60 whose sum is 11. The two factors are 15 and -4.

Using the factors of -60, we express the linear term of the equation as 15x +-4x.


Take the common factors as 3xx+5-4x+5=0.


Thus, the roots are x=-5 or x=43


Solving quadratic equations using Po-Shen Loh Method:

Example 1. What are the roots of 5x2+18x-8=0?


For cases like ax2+bx+c=0 where a1, we divide each term of both sides of the equation by a.



The two numbers whose sum is -185 are -95 and -95

The two roots of the equation is written as x1=-95-u and x2=-95+u

Using the idea of product of -85, we have -95-u-95+u=-85

Solving for u yields ±115. Use either u=-115 or u=115



Solved Examples (Completing the Square and Quadratic Formula)

Example 1. Solve the equation x2-4x-96=0 using completing the square or quadratic formula.

Solution 1. Using Completing the Square

Rewrite the equation as x2-4x=96.

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 x=12 or x=-8


Solution 2. Using the Quadratic Formula

Use the formula x=-b±b2-4ac2a. By direct substitution of a=1, b=-4, and c=-96, we have:


The roots are x=-8 or x=12

Cheat Sheet

  • Use the discriminant D=b2-4ac to determine the nature of the roots of a quadratic equation.
  • If b2-4ac>0 and not a perfect square, then there are two real, irrational roots.
  • If b2-4ac>0 and a perfect square, then there are two unequal real rational roots.
  • If b2-4ac=0, then there is only one real and rational root.
  • if b2-4ac<0, there are no real roots.
  • To formulate a quadratic equation with the given roots x1 and x2, use the model x-x1+x2x+x1x2=0.

Blunder Areas

  • 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 x2+3x+2=0 is written as x+2x+1=0, it is a common mistake to write the roots as x=2 or x=1. The roots must be x=-2 or x=-1.
  • 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: x=-2 or x=-1. It is incorrect to write it as x=-2 and x=-1.
  • If you are asked to write the solution set of a quadratic equation, then write it as x1, x2.
  • Be careful in dealing with signs of the coefficients of ax2+bx+c=0 when using the quadratic formula.