Lesson : Creating and Solving Quadratic Equations in One Variale Prerequisite Skills This lesson requires the use of the following skills: understanding real numers and complex numers understanding rational numers and irrational numers applying the order of operations Introduction Completing the square can e a long process, and not all quadratic expressions can e factored. Rather than completing the square or factoring, we can use a formula that can e derived from the process of completing the square. This formula, called the quadratic formula, can e used to solve any quadratic equation in standard form, ax + x + c = 0. Key Concepts A quadratic equation in standard form, ax + x + c = 0, can e solved for x y using the ac quadratic formula: x = ±. a Solutions of quadratic equations are also called roots. The expression under the radical, ac, is called the discriminant. The discriminant tells us the numer and type of solutions for the equation. Discriminant Numer and type of solutions Negative Two complex solutions 0 One real, rational solution Positive and a perfect square Two real, rational solutions Positive and not a perfect square Two real, irrational solutions Common Errors/Misconceptions not setting the quadratic equation equal to 0 efore determining the values of a,, and c forgetting to use ± for prolems with two solutions forgetting to change the sign of dividing y a or y instead of y a not correctly following the order of operations not fully simplifying solutions U-78
Lesson : Creating and Solving Quadratic Equations in One Variale Guided Practice.. Example 1 Given the standard form of a quadratic equation, ax + x + c, derive the quadratic formula y completing the square. 1. Begin with a quadratic equation in standard form. ax + x + c = 0. Sutract c from oth sides. ax + x = c 3. Divide oth sides y a. x a x c + = a. Complete the square. Add the square of half of the coefficient of the x-term to oth sides to complete the square. Add a to oth sides to complete the square. x + a x + c a = + a a U-79
Lesson : Creating and Solving Quadratic Equations in One Variale. Write the left side as a inomial squared and simplify the right side. c a x + a = + a ac x + a = + a a ac x + a = a 6. Take the square root of oth sides and simplify the right side. x + =± a x + =± a 7. Sutract a x = ± a ac a ac a from oth sides to solve for x. ac a 8. The quadratic formula can e written as two fractions, as in step 7. However, the fractions are often comined. ac a U-80
Lesson : Creating and Solving Quadratic Equations in One Variale Example Use the discriminant of 3x x + 1 = 0 to identify the numer and type of solutions. 1. Determine a,, and c. a = 3, =, and c = 1. Sustitute the values for a,, and c into the formula for the discriminant, ac. ac = ( ) (3)(1) = 1 = 13 The discriminant of 3x x + 1 = 0 is 13. 3. Use what you know aout the discriminant to determine the numer and type of solutions for the quadratic equation. The discriminant, 13, is positive, ut it is not a perfect square. Therefore, there will e two real, irrational solutions. U-81
Lesson : Creating and Solving Quadratic Equations in One Variale Example 3 Solve x x = 1 using the quadratic formula. 1. Write the quadratic in standard form. x x = 1 Original equation x x 1 = 0 Sutract 1 from oth sides.. Determine the values of a,, and c. a =, =, and c = 1 3. Sustitute the values of a,, and c into the quadratic formula. ac a x = ( ) ± ( ) ( )( 1 ) ( ) x = ± + 96 x = ± 11 x = ± 11 Quadratic formula Sustitute values for a,, and c. Simplify.. Determine the solution(s). Since the discriminant, 11, is positive and a perfect square, there are two real, rational solutions. Write the fraction from step 3 as two fractions and simplify. x = + 11 16 = = x = 11 = 6 = 3 x = x = 3 The solutions to the equation x x = 1 are x = or x = 3. U-8
Lesson : Creating and Solving Quadratic Equations in One Variale Example Solve x = x 1 using the quadratic formula. 1. Put the quadratic in standard form. x = x 1 x x = 1 x x + 1 = 0 Original equation Sutract x from oth sides. Add 1 to oth sides.. Determine the values of a,, and c. a = 1, =, and c = 1 3. Sustitute the values of a,, and c into the quadratic formula. ac a x = ( ) ± ( ) 1 ()() 1 1 () x = ± x = ± 0 x = ± 0 Quadratic formula Sustitute values for a,, and c. Simplify.. Determine the solution(s). Since the discriminant is 0, there is one real, rational solution. x = ± 0 = = 1 x = 1 The solution to the equation x = x 1 is x = 1. U-83
Lesson : Creating and Solving Quadratic Equations in One Variale Example Solve x + x + 3 = 0 using the quadratic formula. 1. The quadratic is already in standard form. Determine the values of a,, and c. a =, =, and c = 3. Sustitute these values into the quadratic formula. ac a x = ( ) ± ( ) ( )() 3 ( ) x = ± 60 x = ± 6 Quadratic formula Sustitute values for a,, and c. Simplify. 3. Determine the solution(s). Since the discriminant is negative, there are two complex solutions. Simplify to determine the solutions. x = ± 6 x = ± 1 1 i x = ± 1 x = 1±i 1 The solutions to the equation x + x + 3 = 0 are x = 1±i 1. U-8