1. 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

2. 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 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

3. 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± −(4×2×−20 2×2
Pair up your answers…
𝑊ℎ𝑒𝑛 𝑥=−5, 𝑦=−2 𝑊ℎ𝑒𝑛 𝑥=2, 𝑦=5
𝑥=−5 𝑜𝑟 2

4. 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

5. 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

6. 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

7. 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

8. 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