3.03 Completing the square
Explain how to find the number to add to x^{2} - 5 x to make it a perfect square trinomial.
Consider the expression x^{2} + 6 x.
Does adding 36 to the expression make it a perfect square?
What should be added to x^{2} + 6 x to make it a perfect square?
Consider the equation x^{2} - 3 x = 6. What should be added to the equation in order to complete the square?
Complete the following expressions so they form a perfect square:
x^{2} + 10 x+⬚
x^{2} - 2 x + ⬚
x^{2}-⬚x+16
x^{2} - x + ⬚
x^{2} + m x + ⬚
x^{2} - \dfrac{4}{5} x + ⬚
Complete the following perfect squares:
x^{2} + 4 x + ⬚ = \left(x + ⬚\right)^{2}
x^{2} - 5 x + ⬚ = \left(x - ⬚\right)^{2}
x^{2}-\dfrac{7}{4} x+⬚=(x-⬚)^2
Rewrite the following in the form \left(x + b\right)^{2} + c using the method of completing the square:
x^{2} + 12 x
x^{2} - 12 x
x^{2} + 2 x + 2
x^{2} - 14 x + 56
x^{2} - 6 x + 5
x^{2} + 3 x + 6
x^{2} - 7 x + 14
Factorise the quadratics using completing the square:
y = x^{2} + 56 x + 159
y = x^{2} - 24 x + 63
Factorise the quadratics using the method of completing the square to get them into the form:y = \left(x - a + \sqrt{b}\right) \left(x - a - \sqrt{b}\right)
y = x^{2} - 6 x + 7
y = x^{2} + 8 x + 14
Using the method of completing the square, rewrite 4 x^{2} - 36 x + 79 in the form c\left(\left(x + a\right)^2 + b\right).
Rewrite the following in the form a\left(x + b\right)^2 + c using the method of completing the square:
3 x^{2} + 9 x + 8
4 x^{2} - 11 x + 7
Use completing the square to factorise the quadratic y = 2 x^{2} + 25 x + 23.
Consider the equation x^{2} + 24 x = 10. To solve the equation by completing the square, state whether each of the following lines of working could be used:
Solve the following quadratic equations:
\left(x + 6\right)^{2} = 4
\left( 4 x - 4\right)^{2} = 4
2 \left(x - 3\right)^{2} = 8
\left(x + 2\right)^{2} = 20
\left(x + 5\right)^{2} - 2 = 15
\left(x + 9\right)^{2} = \dfrac{15}{2}
Solve the following quadratic equations by completing the square:
x^{2} + 18 x + 32 = 0
x^{2} - 6 x + 8 = 0
x^{2} - 9 x + 8 = 0
2 x^{2} - 12 x - 32 = 0
4 x^{2} + 11 x + 7 = 0
x^{2} - 2 x - 32 = 0
Solve the following quadratic equations by completing the square. Leave your answers in surd form:
x^{2} + 24 x + 5 = 0
x^{2} + 11 x + 5 = 0
6 x^{2} + 48 x + 24 = 0
x^{2} - 7 x + 8 = 0
x^{2} + \dfrac{x}{3} - 3 = 0
5 x^{2} + 55 x + 3 = 0
A circle has equation \left(x + 2\right)^{2} + \left(y - 4\right)^{2} = 41. Find the x-intercepts of the circle.
Find the value(s) of k that will make x^{2} + k x + 16 a perfect square trinomial.
The revenue y (in millions of dollars) of a company x years after it first started is modelled by y = 12.5 x^{2} - 64 x + 135
By completing the square, use this equation to predict the number of years x after which the revenue of the company will reach \$1167 million.