Methods for Solving Quadratics – Review, Easy
Methods for Solving Quadratics – Review, Hard
Solve each of the following by factorisation:
2𝑥 2 − 3𝑥 − 2 = 0
Solve each of the following by factorisation:
4𝑥 2 − 4𝑥 − 15 = 0
3𝑥 2 − 19𝑥 − 14 = 0
Solve each of the following by completing the square:
4𝑥 2 + 16𝑥 + 7 = 0
6𝑥 2 − 11𝑥 − 10 = 0
Solve each of the following by completing the square:
2𝑥 2 + 16𝑥 + 14 = 0
Solve each of the following by using the formula 𝑥 =
20𝑥 2 + 7𝑥 − 6 = 0
𝑥 2 + 7𝑥 + 12 = 0
−𝑏±√𝑏2 −4𝑎𝑐
:
2𝑎
Solve each of the following by using the formula 𝑥 =
12𝑥 2 − 29𝑥 − 8 = 0
𝑐
𝑎
6𝑥 2 + 11𝑥 + 4 = 0
24𝑥 2 − 41 − 35 = 0
𝑏
𝑎
Solve each of the following by using Vieta’s formulae 𝑥1 𝑥2 = , 𝑥1 + 𝑥2 = − .
−𝑏±√𝑏2 −4𝑎𝑐
:
2𝑎
16𝑥 2 + 42𝑥 + 27 = 0
𝑐
𝑎
𝑏
𝑎
Solve each of the following by using Vieta’s formulae 𝑥1 𝑥2 = , 𝑥1 + 𝑥2 = − .
𝑥 2 − 3𝑥 − 10 = 0
2𝑥 2 + 2𝑥 − 12 = 0
𝑥 2 − 2𝑥 − 99 = 0
2𝑥 2 + 19𝑥 + 35 = 0
2𝑥 2 − 15𝑥 − 8 = 0
6𝑥 2 + 17𝑥 + 5 = 0
10𝑥 2 + 55𝑥 + 60 = 0
18𝑥 2 + 15𝑥 − 168 = 0
Applications of Quadratic Equations – Easy
1. A number squared added to itself makes 90. What was the number?
2. The squares of three consecutive positive integers have sum 302. What
are the three integers?
3. A piece of string forms a rectangle with perimeter 26m and area 30 m2.
What is its length and width?
4. One integer is 4 greater than another, and have a product of 221. If both
integers are negative, what are their values?
5. There is a square piece of fabric that is white with two blue stripes. Both
blue stripes are parallel and run from one end of the fabric to the other.
Both blue stripes are 1 cm in width. The area of the white area is 35cm2.
Calculate the length of one side of the square.
6. The length of a rectangular piece of paper is 4 cm longer than its width.
From all four corners, a square of area 9 cm2 is cut and the piece of paper
is folded to produce an open-topped cuboid with volume 96cm3.
Calculate the length (not width) of the piece of paper (before folded).
Why must the width be greater than 6 cm?
Applications of Quadratic Equations – Hard
1. The squares of three consecutive positive integers have sum 509.
Write an equation representing this in the form 𝑎𝑥 2 + 𝑏𝑥 − 𝑐 = 0
where 𝑎, 𝑏, 𝑐 ∈ ℤ and solve the equation by using the formula 𝑥 =
−𝑏±√𝑏2 −4𝑎𝑐
.
2𝑎
2. One integer 𝑥, is three less than another. Together they have a
product of 378. Write an equation in terms of 𝑥 and solve it by
completing the square.
3. A square ABCD has area 256 𝑐𝑚2 with two points P and Q on ̅̅̅̅
𝐴𝐵 and
̅̅̅̅
̅̅̅̅
𝐴𝐷 respectively where AQ = PB. Points P and Q form 𝑃𝑄 and
complete a right-angled triangle ⊿APQ with area 9 𝑐𝑚2 . State length
AP by creating a statement in the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 where
𝑎, 𝑏, 𝑐 ∈ ℤ and solve by using the formula 𝑥 =
−𝑏±√𝑏2 −4𝑎𝑐
.
2𝑎
4. A right-angled isosceles triangle ⊿ABC with right angle at point B has
̅̅̅̅ and 𝐵𝐶
̅̅̅̅ respectively
area 32 𝑐𝑚2 with two points P and Q on 𝐴𝐵
̅̅̅̅ and complete a trapezium
where AP = CQ. Points P and Q form 𝑃𝑄
2
APQC with area 14 𝑐𝑚 . State distance AP by using the formula 𝑥 =
−𝑏±√𝑏2 −4𝑎𝑐
.
2𝑎