1.09 Completing the square
Find the missing coefficient or term so that the following expressions form a perfect square:
x^{2} + 10 x+⬚
x^{2}-⬚x+16
x^{2}-⬚x+121
For each of the following expressions, determine the value of k to make the expression a perfect square:
Complete the following perfect squares:
\left(x + ⬚\right)^{2} = x^{2} + 20 x + ⬚
\left(x - ⬚\right)^{2} = x^{2} - \dfrac{4}{3} x + ⬚
x^{2} + 4 x + ⬚ = \left(x + ⬚\right)^{2}
x^{2} - 5 x + ⬚ = \left(x - ⬚\right)^{2}
x^{2}-\dfrac{7}{4} x+⬚=\left(x-⬚\right)^2
\left(x - ⬚\right)^{2} = x^{2} - \dfrac{3}{2} x + ⬚
Rewrite the following quadratics in the form \left(x + b\right)^{2} + c using the method of completing the square:
x^{2} + 18 x
x^{2} - 8 x
x^{2} + 10 x + 31
x^{2} + 14 x + 47
x^{2} - 10 x + 30
x^{2} - 18 x + 77
x^{2} + 9 x + 16
x^{2} - 7 x + 15
Factorise:
\left( x - 3 \right)^{2} - 1
\left( x + 4 \right)^{2} - 9
\left( x + 5 \right)^{2} - 49
\left( x - 1 \right)^{2} - 3
Factorise the following quadratics using the method of completing the square:
x^{2} + 24 x + 63
x^{2} - 20 x + 19
x^{2} + 42 x + 185
x^{2} - 28 x + 115
x^{2} + 11 x + 10
x^{2} - 11 x + 30
\left(x + 3\right) \left(x + 19\right) - 17
Find the centre and radius of the following circles:
Complete the following statements:
3 x^{2} + 6 x - 8=⬚\left(x^2 + 2x\right)-8
2 x^{2} - 10 x+1=⬚\left(x^2 - 5x\right)+1
4 x^{2} +12 x - 6=⬚\left(x^2 + ⬚\right)-6
6 x^{2} + 24x - 7=⬚\left(x^2 + ⬚\right)-7
Complete the working to rewrite the following in terms of a\left(x + b\right)^2 + c by completing the square:
Rewrite the following in the form a\left(x + b\right)^2 + c by completing the square:
3 x^{2} + 33 x + 88
5 x^{2} + 5 x + 1
Rewrite the following in the form a\left(\left(x + b\right)^2 + c\right) by completing the square:
Factorise:
3\left( x - 1 \right)^{2} - 12
2\left( x + 4 \right)^{2} - 50
4\left( x + 2 \right)^{2} - 1
-2\left( x - 9 \right)^{2} +32
-4\left( x + 2 \right)^{2} + 16
9\left( x + 2 \right)^{2} - 4
4\left( x + 3 \right)^{2} - 5
2\left( x - 1 \right)^{2} - 3
Factorise the following by completing the square to write in the form y = c \left(x + a\right) \left(x + b\right):
Factorise 2 x^{2} + 11 x + 9 by completing the square to write in the form \left(kx + a\right) \left(x + b\right).
A cube has a surface area of 6 x^{2} + 36 x + 54. What is a length of one of the sides?