Quadratic Simultaneous Equations Grade 9 Advanced Algebra Quadratic Simultaneous Equations Lesson Objective: to recap how to solve linear simultaneous equations and how to solve simultaneous equations when one of them is quadratic

Quadratic Simultaneous Equations Expand the brackets (3+𝑥) 2 = 3+𝑥 3+𝑥 (3+𝑥) 2 =9+ 𝑥 2 +6𝑥 𝑥 2 + 𝑦 2 =29 𝑦−𝑥=3 quadratic linear Substitute this into here… 𝑥 2 +9+ 𝑥 2 +6𝑥=29 Tidy up the equation a little bit… Rearranging the linear equation… 2𝑥 2 +6𝑥+9=29 𝑦=3+𝑥 Prepare our equation so we can use The Quadratic Formula Substitute it into the quadratic equation 𝑥 2 + (3+𝑥) 2 =29 2𝑥 2 +6𝑥−20=0

Quadratic Simultaneous Equations 2𝑥 2 +6𝑥−20=0 Remember our original equations? 𝑥 2 + 𝑦 2 =29 𝑦−𝑥=3 Substitute x = 5 or -2 into the second equation (it’s a lot easier!) a = 2 b = 6 c = -20 𝑦=−2 𝑜𝑟 5 𝑥= −6± 6 2 −(4×2×−20 2×2 Pair up your answers… 𝑊ℎ𝑒𝑛 𝑥=−5, 𝑦=−2 𝑊ℎ𝑒𝑛 𝑥=2, 𝑦=5 𝑥=−5 𝑜𝑟 2

Independent Task 𝑥 2 + 𝑦 2 =25 𝑦=𝑥−7 Hints Substitute the linear equation into the quadratic equation Expand the brackets Prepare your equation so you can use The Quadratic Formula Find two values for x Substitute your two values for x into the linear equation to find two values for y Pair up your answers

Quadratic Simultaneous Equations 𝑥 2 + 𝑦 2 =25 𝑦=𝑥−7 quadratic Substituting into the quadratic equation linear 𝑥 2 + 𝑥 2 −14𝑥+49=25 Tidy up the equation a little bit… Substitute the linear equation into the quadratic equation 2𝑥 2 −14𝑥+49=25 𝑥 2 + (𝑥−7) 2 =25 Prepare our equation so we can use The Quadratic Formula Expand the brackets (𝑥−7) 2 = 𝑥−7 𝑥−7 (𝑥−7) 2 = 𝑥 2 −14𝑥+49 2𝑥 2 −14𝑥+24=0

Quadratic Simultaneous Equations 2𝑥 2 −14𝑥+24=0 Going back to our original equations… 𝑥 2 + 𝑦 2 =25 𝑦=𝑥−7 Substitute x = 3 or 4 into the linear equation a = 2 b = -14 c = 24 𝑦=−4 𝑜𝑟 −3 𝑥= −(−14)± (−14) 2 −(4×2×24) 2×2 Pair up your answers… 𝑊ℎ𝑒𝑛 𝑥=3 𝑦=−4 𝑊ℎ𝑒𝑛 𝑥=4, 𝑦=−3 𝑥=3 𝑜𝑟 4

Quadratic Simultaneous Equations 𝑥 2 + 𝑦 2 =18 𝑦−2𝑥=3 𝑥 2 + 𝑦 2 =40 x+𝑦=4 𝑥 2 + 𝑦 2 =5 𝑦=3𝑥+1 𝑥 2 + 𝑦 2 =13 x−2𝑦=1

Quadratic Simultaneous Equations Solutions 𝑥=−3, 𝑦=−3 𝑥=0.6, 𝑦=4.2 𝑥=6, 𝑦=−2 𝑥=−2, 𝑦=6 𝑥=0.4, 𝑦=2.2 𝑥=−1,𝑦=−2 𝑥=−3, 𝑦=−2 𝑥=3.4, 𝑦=1.2