QUADRATICS
OBJECTIVES
 By the end of the lesson, you should be able to:
 Carry out the completing the square process for a quadratic function and use it to locate the vertex of the graph
 Solve quadratic equations and inequalities in one unknown
 Solve more complex quadratic equations
 Find and use the discriminant of a quadratic polynomial to determine the nature of the roots of the polynomial
 Solve by substitution a pair of simultaneous equation with one being linear and the other quadratic
BRAINSTORM
 Solve the quadratic equation 𝑥 2 − 3𝑥 = 18 by factorization

𝑥−3 𝑥+6 =0

𝑥 = 3 𝑜𝑟 𝑥 = −6
GRAPHS OF QUADRATIC FUNCTIONS
 The sketch of a graph of the form 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 +
𝑐 where a,b and c are constants and 𝑎 ≠ 0 is given
NOTE THAT THE COORDINATE OF THE VERTEX OF THE CURVES
ARE −𝒑, 𝒒 AND CAN BE FOUND BY COMPLETING THE SQUARE
METHOD.
COMPLETING THE SQUARE METHOD
 A quadratic polynomial of the form 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 can be written in form 𝒂 𝒙 + 𝒑
completing the square with p , q as constants.
 This method deals with dividing the coefficient of x by 2
𝟐
+ 𝐪 by the method of
EXAMPLE 1
 Express 𝑦 = 𝑥 2 − 6𝑥 + 10 in the form
𝟐
+ 𝐪. Sketch the curve, stating the
coordinates of the vertex.
 y= 𝒙+𝒑

𝑥−3
2
+1
EXAMPLE 2
 Express 𝑥 2 + 2𝑥 − 5 in the form 𝒂 𝒙 + 𝒑
𝟐
+𝐪
where p and q are constants and find the maximum
value

𝑥+1
2
−6
EXAMPLE 3
 Express 6 − 8𝑥 − 𝑥 2 in the form 𝒂 𝒙 + 𝒑
where p and q are constants.
 − 𝑥+4
2
+ 22
𝟐
+𝐪
ACTIVITY
 Express 6𝑥 2 + 12𝑥 − 3 in the form 𝒂 𝒙 + 𝒑
𝟐
+𝐪
where p and q are constants and sketch stating the
coordinates of the vertex
 6 𝑥+1
2
−9
−1, −9
EXAMPLE 4
 A sheep pen is in the shape of a rectangle. One of
the sides of the pen is a wall. A farmer puts fencing
on the other three sides of the pen. The two sides
that touch the wall are each xm. He uses 40m of
fencing. Find the maximum area of the pen.
GROUP ACTIVITY
 Faisal is x years old. Faisal has a brother called Omar.
The sum of the two boys’ ages is 20years.
 Express the product of their ages in the form 𝑦 =
𝑎 𝑥−𝑏
2
+𝑐
 How old must Faisal be to make the product of their
ages a maximum?
 - 𝑥 − 10
 10years
2
+ 100
SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE
 Example 1
 Solve 2𝑥 2 − 8𝑥 + 1 = 0 by completing the square
method. Leave your answer in surd form
 𝑥 =2±
7
2
ACTIVITY
 Solve 9 + 2𝑥 − 3𝑥 2 = 0 by completing the square
method. Leave your answer in surd form
 𝑥=
1±2 7
3
SOLVING QUADRATIC INEQUALITIES BY COMPLETING THE SQUARE
EXAMPLE1
 Solve the inequality 3𝑥 2 + 24𝑥 + 2 < 0 by
completing the square. Leave your answer in surd
form
EXAMPLE 2
 Solve the inequality 𝑥 2 − 6𝑥 − 3 ≥ 0 by completing
the square. Leave your answer in surd form
PAIR ACTIVITY
 Solve the inequality 𝑥 2 − 2𝑥 − 1 > 0 by completing
the square. Leave your answer in surd form
SOLVING QUADRATIC EQUATIONS USING THE FORMULA
EXAMPLE 1
 Solve the inequality 3 − 5𝑥 − 2𝑥 2 ≥ 0 by using the
formula
ACTIVITY
 Solve the inequality 𝑥 2 < 1 − 𝑥 by using the formula.
SOLVING COMPLEX QUADRATIC EQUATIONS
 We could use the idea of substitution to reduce some complex equation to a quadratic form and then solve to
find the variable
EXAMPLE 1
 Solve 𝑥 4 − 5𝑥 2 + 4 = 0 using the formula
EXAMPLE 2
 Solve 5𝑥 4 − 20𝑥 2 = 1 by completing the square.
Give your answer to 3 significant figures
ACTIVITY
 Solve 6𝑥 8 + 6 = 13𝑥 4
THE DISCRIMINANT OF A QUADRATIC EQUATION
 The roots of a quadratic equation are the values got
when the equation has been solved.
 The roots tell us where the quadratic graph crosses
the x-axis
 The discriminant is the value 𝒃𝟐 − 𝟒𝒂𝒄 in the
quadratic formula
 If 𝒃𝟐 − 𝟒𝒂𝒄 > 𝟎, there are two distinct real roots
 𝒃𝟐 − 𝟒𝒂𝒄 = 𝟎, then there equal roots
 𝒃𝟐 − 𝟒𝒂𝒄 < 𝟎 , then there are no real roots
EXAMPLE 1
 Work out whether each of these quadratic equations
has two distinct roots, equal roots or no real roots.
 3𝑥 2 + 𝑥 − 6 = 0
 𝑥 2 − 3𝑥 + 5 = 0
 25𝑥 2 + 20𝑥 + 4 = 0
ACTIVITY
 Find the relationship between p and q, if the equation
𝑝𝑥 2 + 3𝑞𝑥 + 9 = 0 has equal roots
 𝑞 2 = 4𝑝
SOLVING SIMULTANEOUS EQUATIONS
 We already know how to solve simultaneous
equations when both equations are linear
 We will solve simultaneous equations where one is
linear and the other quadratic.
 We will either get two distinct roots, one repeated
root or no real roots
EXAMPLE 1
 Solve simultaneously 𝑦 = 𝑥 2 − 3𝑥 − 1 𝑎𝑛𝑑 𝑦 =
2𝑥 − 7

3, −1 𝑎𝑛𝑑 2, −3
EXAMPLE 2
 Show that there are no real roots for the
simultaneous equations 𝑦 = 𝑥 2 − 2𝑥 − 1 𝑎𝑛𝑑 𝑦 =
𝑥−5
 Hint: use the discriminant
EXAMPLE 3
 Solve simultaneously 𝑦 2 + 𝑥𝑦 + 4𝑥 = 7 𝑎𝑛𝑑 𝑥 −
𝑦=3

1 −5
,
2 2
𝑎𝑛𝑑 2, −1
ACTIVITY
 Solve the simultaneous equation
 𝑦=𝑥+1
 𝑦 = 𝑥2 − 1

−1,0 𝑎𝑛𝑑 2,3
THANK YOU