International Studies Charter School Bell Ringer Mrs. Rivas International Studies Charter School Factor the following expression. 𝒂)− 𝒙 𝟐 +𝟑𝒙+𝟏𝟖 𝒃)−𝟑 𝒙 𝟐 +𝒙+𝟏𝟒 − 𝒙 𝟐 −𝟑𝒙−𝟏𝟖 − 𝟑𝒙 𝟐 −𝒙−𝟏𝟒 𝟑𝒙 𝒙 −𝟔 −𝟔𝒙 𝟔𝒙 𝟑𝒙 −𝟕 −𝟕𝒙 𝒙 + 𝟑 𝒙 + 𝟐 𝟑𝒙−𝟔𝒙=−𝟑𝒙 𝟔𝒙−𝟕𝒙=−𝒙 − 𝒙−𝟔 𝒙+𝟑 − 𝟑𝒙−𝟕 𝒙+𝟐

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square Objective: To solve equations by completing the square and to rewrite functions by completing the square Essential Understanding Completing a perfect square trinomial allows you to factor the completed trinomial as the square of a binomial.  You can solve an equation that contains a perfect square by finding square roots. The simplest of this type of equations has the form 𝒂 𝒙 𝟐 =𝒄

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square −𝟏𝟎 −𝟏𝟎 +𝟓 +𝟓 𝟒 𝒙 𝟐 =𝟑𝟔 𝟑 𝒙 𝟐 =𝟑𝟎 𝟒 𝟒 𝟑 𝟑 𝒙 𝟐 =𝟗 𝒙 𝟐 =𝟏𝟎 𝒙= 𝟗 𝒙=± 𝟏𝟎 𝒙=±𝟑

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square 𝒙=± 𝟓 𝒙=± 𝟐

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square 𝟏.𝟔 𝟏 = ? 𝟒𝟐 𝒊𝒏 𝟏.𝟔 𝟏 = 𝟐𝟖𝟐𝟐.𝟒 𝒊𝒏 𝟐 𝒙 𝟐 𝟏.𝟔 𝒙 𝟐 =𝟐𝟖𝟐𝟐.𝟒 𝒊𝒏 𝟐 𝟏.𝟔 𝟏 = 𝟔𝟐.𝟐 𝒊𝒏 𝟒𝟐 𝒊𝒏 𝒙 𝟐 =𝟏𝟕𝟔𝟒 𝒊𝒏 𝟐 𝒙 𝟐 = 𝟏𝟕𝟔𝟒 𝒊𝒏 𝟐 𝟒𝟐 𝒊𝒏 𝒃𝒚 𝟔𝟕.𝟐 𝒊𝒏 𝒙=𝟒𝟐 𝒊𝒏

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square Completing the Square You can for a perfect square trinomial from 𝒙 𝟐 +𝒃𝒙 by adding 𝒃 𝟐 𝟐 𝒙 𝟐 +𝒃𝒙+ 𝒃 𝟐 𝟐 = 𝒙+ 𝒃 𝟐 𝟐

Completing the Square Mrs. Rivas Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square Solving an equation by completing the square Step 1.  Rewrite the equation in the form 𝒙 𝟐 +𝒃𝒙=𝒄. To do this, get all terms with the variable on one side of the equation and the constant on the other side. Divide all the terms of the equation by the coefficient of 𝑥 2 if it is not 1. Step 2. Complete the square by adding 𝒃 𝟐 𝟐 to each side of the equation. Step 3. Factor the trinomial. Step 4. Find the square root. Step 5. Solve for 𝑥.

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square

International Studies Charter School Remember c is the y-intercept Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square Step 2 Find 𝑏 2 2 𝑥 2 +4𝑥 + _____=6+____ Step 1 move c to the other side 𝑥 2 +4𝑥 +𝟒=6+𝟒 = 4 2 2 Step 3 Factor the perfect square trinomial 𝑥 2 +4𝑥 +𝟒=10 𝑥+2 2 =10 Step 4 move c back to the left =𝟒 𝑦= 𝑥+2 2 −10 Remember c is the y-intercept 𝒗𝒆𝒓𝒕𝒆𝒙 −𝟐,−𝟏𝟎 𝒚−𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 𝟎,−𝟔 𝒚=𝒂 𝒙 𝟐 +𝒃𝒙+𝒄

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square 𝑥 2 +3𝑥 + _____=6+____ 𝑥 2 +3𝑥+ 𝟗 𝟒 =6+ 𝟗 𝟒 = 3 2 2 𝑥+ 3 2 2 = 𝟑𝟑 𝟒 = 𝟗 𝟒 𝑦= 𝑥+ 3 2 2 − 𝟑𝟑 𝟒 𝒗𝒆𝒓𝒕𝒆𝒙 − 𝟑 𝟐 ,− 𝟑𝟑 𝟒 𝒚−𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 𝟎,−𝟔

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square

Completing the Square Mrs. Rivas 2 𝑥 2 −𝑥+3=𝑥+9 −𝑥−9 −𝑥−9 Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square Ignore the 2 completely since 2 is not and answer. 2 𝑥 2 −𝑥+3=𝑥+9 −𝑥−9 −𝑥−9 𝑥− 1 2 2 = 13 4 2 𝑥 2 −2𝑥−6=0 2 𝑥 2 −𝑥−3 =0 𝑥− 1 2 2 = 13 4 𝑥 2 −𝑥−3=0 𝑥 2 −𝑥=3 𝑥− 1 2 = 13 2 𝑥 2 −𝑥+ −1 2 2 =3 + −1 2 2 𝑥 2 −𝑥+ 1 4 =3+ 1 4 𝒙= 𝟏± 𝟏𝟑 𝟐

International Studies Charter School Section 4-6 Mrs. Rivas International Studies Charter School Completing the Square 2 𝑥 2 −𝑥+3=𝑥+9 𝑥= 2± 52 4 −𝑥−9 −𝑥−9 2 𝑥 2 −2𝑥−6=0 𝑥= 2± 2∙2∙13 4 𝑎 𝑏 𝑐 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 𝑥= 2±2 13 4 𝑥= −(−2)± (−2) 2 −4(2)(−6) 2(2) 𝑥= 𝟏± 𝟏𝟑 𝟐 𝑥= 2± 4+48 4

International Studies Charter School Section 4-7 Mrs. Rivas International Studies Charter School Quadratic Formula 𝒙= −𝒃± 𝒃²−𝟒𝒂𝒄 𝟐𝒂

International Studies Charter School Section 4-7 Mrs. Rivas International Studies Charter School Quadratic Formula 𝟐𝒙²−𝒙=𝟒 Step # 1: Write the equation in standard form. 𝒚=𝒂𝒙²+𝒃𝒙+𝒄 𝟐𝒙²−𝒙−𝟒=𝟒−𝟒 𝟐𝒙²−𝒙−𝟒=𝟎 Step # 2: Find the values of 𝒂, 𝒃 and 𝒄. 𝒂=𝟐 𝒃=−𝟏 𝒄=−𝟒 𝒙= −(−𝟏)± −𝟏 2 −𝟒(𝟐)(−𝟒) 𝟐(𝟐) 𝒙= −𝒃± 𝒃²−𝟒𝒂𝒄 𝟐𝒂 Step # 3: Write the quadratic formula and substitute 𝒂, 𝒃 and 𝒄. Step # 4: Simplify. 𝒙= 𝟏± 𝟑𝟑 𝟒 Answer: 𝟏+ 𝟑𝟑 𝟒 and 𝟏− 𝟑𝟑 𝟒

International Studies Charter School Section 4-7 Mrs. Rivas International Studies Charter School Quadratic Formula 𝒙²+𝟔𝒙+𝟗=𝟎 Step # 1: Write the equation in standard form. 𝒚=𝒂𝒙²+𝒃𝒙+𝒄 𝒄=𝟗 𝒂=𝟏 𝒃=𝟔 Step # 2: Find the values of 𝒂, 𝒃 and 𝒄. 𝒙= −(𝟔)± 𝟔 2 −𝟒(𝟏)(𝟗) 𝟐(𝟏) 𝒙= −𝒃± 𝒃²−𝟒𝒂𝒄 𝟐𝒂 Step # 3: Write the quadratic formula and substitute 𝒂, 𝒃 and 𝒄. 𝒙= −𝟔± 𝟎 𝟐 Step # 4: Simplify. Answer: 𝒙=−𝟑

International Studies Charter School Section 4-7 Mrs. Rivas International Studies Charter School Quadratic Formula 𝒙= −(𝟒)± 𝟒 2 −𝟒(𝟏)(𝟒) 𝟐(𝟏) 𝒙= −(𝟒)± 𝟒 2 −𝟒(𝟏)(−𝟑) 𝟐(𝟏) 𝒙= −𝟒± 𝟐𝟖 𝟐 𝒙= −𝟒± 𝟎 𝟐 𝒙=−𝟐 𝒙= −𝟒+ 𝟐𝟖 𝟐 𝐚𝐧𝐝 𝒙== −𝟒− 𝟐𝟖 𝟐

International Studies Charter School Section 4-7 Mrs. Rivas International Studies Charter School Quadratic Formula

International Studies Charter School Section 4-7 Mrs. Rivas International Studies Charter School Quadratic Formula

International Studies Charter School Section 4-7 Mrs. Rivas International Studies Charter School Quadratic Formula

International Studies Charter School Section 4-7 Mrs. Rivas International Studies Charter School Quadratic Formula

International Studies Charter School Mrs. Rivas International Studies Charter School Pg. 237 to 239 # 13, 17, 23, 25, 27, 29, 31, 35, 39, 40, 45, 47, 51, 86, 87, 91, 92 Pg. 245 11, 15, 19, 21, 25, 27, 29, 35, 36