Solving Quadratic Equations Using the Quadratic Formula MA.912.A.7.2 Solve quadratic equations over the real numbers by factoring and by using the quadratic formula.

The Quadratic Formula The solutions of a quadratic equation written in Standard Form, ax2 + bx + c = 0, can be found by Using the Quadratic Formula. Have students sing song together. Click on the link below to view a song to help you memorize it. http://www.regentsprep.org/Regents/math/algtrig/ATE3/quadsongs.htm

Deriving the Quadratic Formula by Completing the Square. Divide both sides by “a”. Subtract constant from both sides.

Deriving the Quadratic Formula by Completing the Square. Complete The Square Factor the Perfect Square Trinomial Simplify expression on the left side by finding the LCD

Deriving the Quadratic Formula by Completing the Square. Take the square root of both sides Solve absolute value/ Simplify radical

Deriving the Quadratic Formula by Completing the Square. Isolate x Simplify Congratulations! You have derived The Quadratic Formula

#1 Solve using the quadratic formula. Review the meaning of the solutions as they relate to the graph of the parabola– the x intercepts of the parabola would be 2 and ⅓. Review other characteristics of the related parabola– it opens up, the vertex is the minimum, the y-intercept is 2.

Graph Clink on link for graphing calculator. http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

#1 Solve by factoring Discuss with students why they might want to choose factoring instead of the quadratic formula.

#2 Solve by factoring This quadratic is Prime (will not factor), Give students time to try to factor. Ask them if they can think of another way to solve since factoring in not possible– they can use the quadratic formula. This quadratic is Prime (will not factor), The Quadratic Formula must be used!

#2 Solve using the quadratic formula. Exact Solution Stress to students that radicals should be left in simplest radical form. This approximate solution has been rounded to the nearest hundredth. Ask Approx Solution

Graph Clink on link for graphing calculator. http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

#3 Solve using the quadratic formula Discuss the fact that there are imaginary solutions for this equation, however in Algebra I we only solve over real numbers. Discuss characteristics of related parabola: “Does this parabola open up or down?”– Up, because a>0. “Does this parabola have any x-intercepts/” – No because there are no Real Solutions to the equation. “Is the vertex above or below the x-axis?”– Above, because it opens up and does not intercept the x-axis. The is not a real number, therefore this equation has ‘NO Real Solution’

Graph Clink on link for graphing calculator. http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

#4 Solve using the quadratic formula Point out the Identity property of addition (adding/subtracting zero, does not change a number). Stress to students that they should always attempt factoring before using the quadratic formula, because factoring is less time consuming and less steps. Review using the zero product property to solve and also rewrite as (x-8)2 = 0 and take the square root of both sides. Would factoring work to solve this equation?

Graph Clink on link for graphing calculator. http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

#5 Solve using the quadratic formula. Exact Solution Stress to students that radicals should be left in simplest radical form. This approximate solution has been rounded to the nearest hundredth. Ask Approx Solution

#5 What if we move everything to the right side? Exact Solution Stress to students that radicals should be left in simplest radical form. This approximate solution has been rounded to the nearest hundredth. Ask Approx Solution

Graph Clink on link for graphing calculator. http://my.hrw.com/math06_07/nsmedia/tools/Graph_Calculator/graphCalc.html

The Discriminant The expression inside the radical in the quadratic formula is called the Discriminant. The discriminant can be used to determine the number of solutions that a quadratic has.

Understanding the discriminant # of real solutions Perfect square 2 real rational solutions Not Perfect 2 real irrational solutions 1 real rational solution Review definition of Rational vs. Irrational. No real solution

#6 Find the discriminant and describe the solutions to the equations. 2 Real Rational Solutions

#7 Find the discriminant and describe the solutions to the equations. No Real Solutions

#8 Find the discriminant and describe the solutions to the equations. 2 Real Rational Solutions

#9 Find the discriminant and describe the solutions to the equations. 1 Real Rational Solution