Algebra 2 – Outcomes Construct a quadratic equation from its roots. Solve systems of equations with one linear and one quadratic equation. Solve linear inequalities. Change the subject of a formula. Solve exponential equations.

Construct a Quadratic Equation Recall a previous quadratic equation 𝑥 2 +5𝑥+4=0: 𝑥+4 𝑥+1 =0 ⇒𝑥=−4 or 𝑥=−1 -4 and -1 are called the roots of the equation. Graphically, where the equation crosses the 𝑥 axis. We can create a quadratic equation from given roots if we work backwards.

Construct a Quadratic Equation e.g. 𝑥=−4 or 𝑥=−1 ⇒𝑥+4=0 or 𝑥+1=0 ⇒ 𝑥+4 𝑥+1 =0 ⇒ 𝑥 2 +𝑥+4𝑥+1=0 ⇒ 𝑥 2 +5𝑥+1=0

Construct a Quadratic Equation e.g. Construct a quadratic equation with each of the following pairs of roots: 𝑥=6 and 𝑥=1 𝑥=4 and 𝑥=10 𝑥=5 and 𝑥=−4 𝑥=−3 and 𝑥=7 𝑥= 1 2 and 𝑥=4 𝑥=− 8 5 and 𝑥= 1 2

Solve Linear-Quadratic Systems Recall solving linear-circle problems in coordinate geometry, for example: 𝑥+𝑦=4⇒𝑥=4−𝑦 𝑥 2 + 𝑦 2 =10 ⇒ 4−𝑦 2 + 𝑦 2 =10 ⇒16−4𝑦−4𝑦+ 𝑦 2 + 𝑦 2 =10 ⇒2 𝑦 2 −8𝑦+6=0 ⇒ 𝑦 2 −4𝑦+3=0 ⇒ 𝑦−3 𝑦−1 =0 ⇒𝑦=3, 1 Rearrange line to 𝑥= or 𝑦= Substitute into circle eqn Simplify and solve quadratic

Solve Linear-Quadratic Systems 𝑥=4−𝑦 𝑦=3, 1 𝑥=4−3=1⇒(1,3) or 𝑥=4−1=3⇒(3,1) Substitute 𝑦 values back into line equation

Solve Linear-Quadratic Systems 2004 OL P1 Q3 Solve for 𝑥 and 𝑦 𝑥+𝑦=1 𝑥 2 + 𝑦 2 =13 2006 OL P1 Q3 Solve for 𝑥 and 𝑦 𝑥−2𝑦=10 𝑥 2 + 𝑦 2 =20 2012 OL P1 Q3 Solve for 𝑥 and 𝑦 𝑥−𝑦+5=0 𝑥 2 + 𝑦 2 =17 Which solution gives the lesser value of 𝑥−2𝑦? Write down this value.

Solve Linear Inequalities (Graphic) Given the function 𝑓 𝑥 =3𝑥−2, Plot a graph of 𝑓 𝑥 in the domain −3≤𝑥≤3. Use your graph to solve the inequality 𝑓(𝑥)≤0. Given the function 𝑔 𝑥 = 4𝑥 3 +1, Plot a graph of 𝑔(𝑥) in the domain −6≤𝑥≤−2. Use your graph to solve the inequality 𝑔 𝑥 <0. Given the functions ℎ 𝑥 =3𝑥−4 and 𝑘 𝑥 =𝑥−2 Plot ℎ(𝑥) and 𝑘(𝑥) in the domain −2≤𝑥≤4. Use your graph to solve the inequality ℎ 𝑥 >𝑘(𝑥).

Solve Linear Inequalities (Algebraic) The rules for solving inequalities are almost the same as the rules for solving equalities. 𝟑<𝟔 3+3<6+3⇒6<9 Addition valid 3−3<6−3⇒0<3 Subtraction valid 3×3<6×3⇒9<18 Multiplication valid 3 3 < 6 3 ⇒1<2 Division valid 3×−3<6×−3⇒−9≮−18 Negative multiplication invalid 3 −3 < 6 −3 ⇒−1≮−2 Negative division invalid

Solve Linear Inequalities (Algebraic) Multiplying or dividing by a negative number makes the inequality invalid. When multiplying or dividing by a negative number, the direction of the inequality must be reversed. e.g. 3×−3<6×−3⇒−9>−18

Solve Linear Inequalities (Algebraic) ⇒3𝑥≤2 ⇒𝑥≤ 2 3 Solve 4𝑥 3 +1<0 ⇒ 4𝑥 3 <−1 ⇒4𝑥<−3 ⇒𝑥<− 3 4

Solve Linear Inequalities (Algebraic) ⇒3𝑥−𝑥>−2+4 ⇒2𝑥>2 ⇒𝑥> 2 2 ⇒𝑥>1

Solve Linear Inequalities (Algebraic) Recall the sets: ℕ - natural positive whole numbers ℤ - integers all whole numbers ℝ - real all numbers Sometimes we must plot our solutions on number lines, which varies with set. e.g. plot 𝑥>−3 for 𝑥∈ℕ, 𝑥∈ℤ, 𝑥∈ℝ on number lines. 𝑥∈ℕ 𝑥∈ℤ 𝑥∈ℝ For 𝑥≥−3, simply add a dot to −3 for 𝑥∈ℤ and 𝑥∈ℝ

Solve Linear Inequalities (Algebraic) e.g. solve each of the following and plot their solutions on an appropriate number line. 𝑥+3<10;𝑥∈ℕ 3𝑥−2≤0;𝑥∈ℤ 4𝑥 3 +1<0;𝑥∈ℝ 3𝑥−4>𝑥−2;𝑥∈ℤ 4𝑞+3≤2 𝑞+3 ;𝑥∈ℝ 4 3 𝑥−6<7𝑥+2;𝑥∈ℤ 7 3𝑥+2 −5 2𝑥−3 >7;𝑥∈ℕ 𝑥−7 3 > 2𝑥−3 2 ;𝑥∈ℕ

Solve Linear Inequalities (Algebraic) 2003 OL P1 Q3 Find the solution set of 5𝑥−3<12, 𝑥∈ℕ 2007 OL P1 Q3 Find the solution set of 4𝑥−15<1, 𝑥∈ℕ 2010 OL P1 Q3 Find the values of 𝑥 which satisfy 2 3+4𝑥 ≤22, 𝑥∈ℕ 2006 OL P1 Q2 Find the smallest natural number 𝑘 such that 2𝑥+4 𝑥+3 +7 2𝑥+4 <20(𝑥+𝑘)

Solve Linear Inequalities (Algebraic) 2005 OL P1 Q3 Find 𝐴, the solution set of 3𝑥−2<4, 𝑥∈ℤ Find 𝐵, the solution set of 1−3𝑥 2 <5, 𝑥∈ℤ List the elements of 𝐴∩𝐵. 2011 OL P1 Q3 Find 𝐴, the solution set of 3𝑥−5<7, 𝑥∈ℤ Find 𝐵, the solution set of −2−3𝑥 4 <1, 𝑥∈ℤ List the elements of 𝐴∩𝐵.

Change the Subject of a Formula When a variable is expressed as an expression of other variables, we call it the subject of the equation. e.g. for 𝑦=𝑚𝑥+𝑐 we call 𝑦 the subject because we are expressing it as a combination of the other variables 𝑚, 𝑥, and 𝑐. e.g. identify the subject of each of the following: 𝑣=𝑢+𝑎𝑡 𝑉=𝜋 𝑟 2 ℎ 𝑏𝑐−𝑑=𝑎 𝑇 2 = 4 𝜋 2 𝑅 3 𝐺𝑀

Change the Subject of a Formula To change the subject of a formula, apply the normal rules of algebra so that the given variable is the new subject: e.g. change the subject of 𝐶= 5 9 (𝐹−32) to 𝐹 𝐶= 5 9 (𝐹−32) Multiply by 9 ⇒9𝐶=5(𝐹−32) Divide by 5 ⇒ 9𝐶 5 =𝐹−32 Add 32 ⇒ 9𝐶 5 +32=𝐹

Change the Subject of a Formula Express 𝑚= 𝑣 𝑢 in terms of 𝑣. Express 𝑃𝑉=𝑛𝑅𝑇 in terms of 𝑅. Express 𝐹=𝑃 1+𝑖 𝑡 in terms of 𝑃. Express 𝐹= 1 4𝜋𝜀 𝑞 1 𝑞 2 𝑑 2 in terms of 𝜀. Express 1 𝑓 = 1 𝑢 + 1 𝑣 in terms of 𝑓. Express 𝑇 2 = 4 𝜋 2 𝑅 3 𝐺𝑀 in terms of 𝜋. Express 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 in terms of 𝑎.

Change the Subject of a Formula Find 𝑐 if 5𝑎+𝑎𝑐=3𝑎 Find 𝑟 if 3 2𝑟−5 =27 Find 𝑥 if 0.5 𝑥−8 =0.2𝑥+11 Find 𝑛 if 9−2𝑛=3 𝑛+2 Find 𝑎 if 𝑎 2 =49 Find 𝑏 if 2𝑏−8 4 =𝑏−3 Find 𝑦 if 0.1𝑦+7 2 = 𝑦 2.5

Solve Problems by Rearranging Consider an animal pen measuring 𝑚 metres long and 𝑛 metres wide. Write down a formula for the perimeter of the pen. If the pen is 5 metres long and 4 metres wide, calculate the perimeter of the pen. If the perimeter of the pen is 200 metres and its length is 10 metres, calculate the width of the pen. If the total area of land is 400 square metres, what is the width of the largest possible square pen?

Solve Problems by Rearranging The acceleration of an object is given by 𝑎= 𝑣−𝑢 𝑡 , where 𝑣 is the final speed, 𝑢 is the starting speed, and 𝑡 is the time taken to accelerate. Write this formula in terms of 𝑣. If a car accelerates from rest at 5 metres per second per second over 4 seconds, what is its final speed? The car then comes to a sudden stop at a red light, stopping in just 2 seconds. What is its acceleration? The distance travelled by the car is given by 𝑠= 𝑢+𝑣 2 ×𝑡. Create a formula for 𝑠 in terms of 𝑢, 𝑎, and 𝑡.

Solve Problems by Rearranging The first €33 800 of annual income for a single person is taxed at 20%. The remainder of their annual earnings is taxed at 40%. Single people are exempt from €1650 of tax per year (called their tax credit). Write a formula for the amount of tax paid by a single person each year. If a single person earns €50 000 in one year, how much tax do they pay? If a single person pays €2 000 in tax in one year, what was their income before tax?

Change the Subject of a Formula 2005 OL P1 Q3 Given that 𝑎𝑥+𝑏=𝑐, express 𝑥 in terms of 𝑎, 𝑏 and 𝑐, where 𝑎≠0. 2008 OL P1 Q3 Given that 𝑎 𝑥+5 =8, express 𝑥 in terms of 𝑎. 2002 OL P1 Q3 Express 𝑏 in terms of 𝑎 and 𝑐 where 8𝑎−5𝑏 𝑏 =𝑐

Solve Exponential Equations Pg 21 of F&T booklet Recall the rules for indices: 𝑎 𝑝 𝑎 𝑞 = 𝑎 𝑝+𝑞 𝑎 𝑝 𝑞 = 𝑎 𝑝−𝑞 𝑎 𝑝 𝑞 = 𝑎 𝑝𝑞 𝑎 0 =1 𝑎 −𝑝 = 1 𝑎 𝑝 𝑎 1 𝑞 = 𝑞 𝑎 𝑎 𝑝 𝑞 = 𝑞 𝑎 𝑝 = 𝑞 𝑎 𝑝 𝑎𝑏 𝑝 = 𝑎 𝑝 𝑏 𝑝 𝑎 𝑏 𝑝 = 𝑎 𝑝 𝑏 𝑝 These formulas can be applied in either direction

Solve Exponential Equations Exponential equations are solved in two parts: Write the equation so that each side is in index form with the same base. Write a new equation from the powers and solve. 3× 3 𝑥 =9 ⇒ 3 1 × 3 𝑥 = 3 2 Write in index form ⇒ 3 𝑥+1 = 3 2 Using 𝑎 𝑝 𝑎 𝑞 = 𝑎 𝑝+𝑞 ⇒𝑥+1=2 Set power = power ⇒𝑥=1

Solve Exponential Equations e.g. Solve 2 𝑥+4 = 4 2𝑥 ⇒ 2 𝑥+4 = 2 2 2𝑥 Using 2 2 =4 ⇒ 2 𝑥+4 = 2 4𝑥 Using 𝑎 𝑝 𝑞 = 𝑎 𝑝𝑞 ⇒𝑥+4=4𝑥 Set power = power ⇒4=3𝑥 ⇒ 4 3 =𝑥

Solve Exponential Equations Solve each of the following: 2 𝑥 =2 2 𝑥+5 = 2 5 3 2𝑥+1 = 3 3 125 𝑥 =5 3 9𝑥−2 =27 64 𝑥+1 = 16 2𝑥+5 81 𝑘+2 = 27 𝑘+4 25 1−2𝑥 = 5 4 Adapted from FHSST grade 10

Solve Exponential Equations 2004 OL P1 Q2 Evaluate 8 1 3 Express 4 1 4 in the form 2 𝑘 . 𝑘∈ℚ Solve for 𝑥 the equation 8 1 3 4 1 4 = 2 5−𝑥 2003 OL P1 Q2 Solve for 𝑥 the equation 25 𝑥 = 5 6−𝑥 2001 OL P1 Q2 Solve each of the following equations for 𝑝 9 𝑝 = 1 3 2 3𝑝−7 = 2 6 − 2 5

Solve Exponential Equations 2007 OL P1 Q2 Find the value of 𝑥 for which 2 𝑥+3 = 4 𝑥 . 2008 OL P1 Q2 Find the value of 𝑥 for which 5 𝑥 3 = 5 6 75 2009 OL P1 Q2 Find the value of 3 6 . Write 27 in the form 3 𝑘 , where 𝑘∈ℕ. Find the value of 𝑥 for which 27× 3 𝑥 = 1 729