Methods for solving quadratic equations
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This document outlines four methods for solving quadratic equations: factoring, principle of square roots, completing the square, and the quadratic formula. It provides examples of how to use each method and explains the key steps involved. Factoring works when the quadratic expression is factorable. The principle of square roots is used when the equation involves a square and constant. Completing the square transforms the equation into perfect square form. The quadratic formula can solve any quadratic equation and provides both real and imaginary solutions.
Methods for solving quadratic equations
- 1. Methods for Solving Quadratic Equations Quadratics equations are of the form 0,02 ≠=++ awherecbxax Quadratics may have two, one, or zero real solutions. 1. FACTORING Set the equation equal to zero. If the quadratic side is factorable, factor, then set each factor equal to zero. Example: 652 −−= xx Move all terms to one side 0652 =++ xx Factor 0)2)(3( =++ xx Set each factor to zero and solve 0203 =+=+ xx 23 −=−= xx 2. PRINCIPLE OF SQUARE ROOTS If the quadratic equation involves a SQUARE and a CONSTANT (no first degree term), position the square on one side and the constant on the other side. Then take the square root of both sides. (Remember, you cannot take the square root of a negative number, so if this process leads to taking the square root of a negative number, there are no real solutions.) Example 1: 0162 =−x Move the constant to the right side 162 =x Take the square root of both sides 162 ±=x 44,4 −==±= xandxmeanswhichx Example 2: 014)3(2 2 =−+x Move the constant to the other side 14)3(2 2 =+x Isolate the square 7)3( 2 =+x (divide both sides by 2) Take the square root of both sides 7)3( 2 ±=+x 73 ±=+x Solve for x 73 ±−=x This represents the exact answer. Decimal approximations can be found using a calculator.
- 2. 3. COMPLETING THE SQUARE If the quadratic equation is of the form 0,02 ≠=++ awherecbxax and the quadratic expression is not factorable, try completing the square. Example: 01162 =−+ xx **Important: If 1≠a , divide all terms by “a” before proceeding to the next steps. Move the constant to the right side 1162 =+ xx Find half of b, which means 2 b : 3 2 6 = Find 93: 2 2 2 = b Add 2 2 b to both sides of the equation 911962 +=++ xx Factor the quadratic side 20)3)(3( =++ xx (which is a perfect square because you just made it that way!) Then write in perfect square form 20)3( 2 =+x Take the square root of both sides 20)3( 2 ±=+x 203 ±=+x Solve for x 523 203 ±−= ±−= x radicaltheSimplifyx This represents the exact answer. Decimal approximations can be found using a calculator. 4. QUADRATIC FORMULA Any quadratic equation of the form 0,02 ≠=++ awherecbxax can be solved for both real and imaginary solutions using the quadratic formula: a acbb x 2 42 −±− = Example: )11,6,1(01162 −====−+ cbaxx Substitute values into the quadratic formula: answerEXACTsimplifiedfinaltheisThisxx radicalthesimplifyxxx 523 2 546 2 806 2 44366 )1(2 )11)(1(466 2 ±−=→ ±− = ±− =→ +±− =→ −−±− =