4 Quadratics and polynomials

Worksheet

1

The equation x^{2} - 144 = 0 has a positive integer solution of x = 12. Find its other solution.

2

Solve the following equations:

a

x^{2} = 2

b

x^{2} = 25

c

x^{2} = 121

d

x^{2} = 294

e

x^{2} - 121 = 0

f

x^{2} - 10 = 15

g

\dfrac{x^{2}}{16} - 2 = 2

h

\dfrac{x^{2}}{25} - 3 = 6

i

\left(x + 3\right)^{2} = 49

j

\left(x - 3\right)^{2} = 64

k

\left(x - 6\right)^{2} = 2

l

\left(2 - x\right)^{2} = 81

m

\left(x - 7\right)^{2} = 81

n

\left(7 - x\right)^{2} = 81

o

\left( 8 x + 9\right)^{2} = 256

p

81 x^{2} - 16 = 0

3

Solve the following equations:

a

x \left(x + 7\right) = 0

b

x \left( 2 x - 9\right) = 0

c

\left( 10 x - 9\right)^{2} = 0

d

\left( 4 x - 9\right)^{2} = 0

e

\left( - 3 + 7 x\right)^{2} = 0

f

\left(x - 4\right) \left(x - 2\right) = 0

g

\left(x - 6\right) \left(x + 7\right) = 0

h

\left( 8 x - 5\right) \left( 3 x - 7\right) = 0

i

\left( 3 x + 8\right) \left( 5 x - 7\right) = 0

j

\left( 3 x - 17\right) \left( 2 x - d\right) = 0

4

Solve the following equations:

a

4 y^{2} = 100

b

25 y^{2} = 36

c

- 3 k^{2} = - 12

d

81 k^{2} + 8 = 24

e

- 25 v^{2} + 64 = 0

f

10 \left(p^{2} - 7\right) = 930

g

4 m \left(m + 5\right) = 0

h

\dfrac{m}{2} \left(m + 5\right) = 0

5

Solve the following equation for x, in terms of a and c. Assume a and c are positive.

a x^{2} - c = 0

6

The equation 4 x^{2} + k x + 16 = 0 has one solution: x = 2. Find the value of the coefficient k.

7

The equation a x^{2} - 32 x - 80=0 has one soulution: x = 4. Find the value of the coefficient of a.

8

The Widget and Trinket Emporium has released the forecast of its revenue over then next year. The revenue R (in dollars) at any point in time t (in months) is described by the equation:R = - \left(t - 12\right)^{2} + 4

When will the revenue be zero?

9

Neville needs a sheet of paper x \,\text{cm} by 13 \, \text{cm} for an origami giraffe. The local origami supply store only sells square sheets of paper.

The lower portion of the image below shows the excess area A of paper that will remain after Neville cuts out the x \,\text{cm} by 13 \,\text{cm} piece. The excess area, in square centimeter, is given by the equation:A = x \left(x - 13\right)

a

At what lengths x will the excess area be zero?

b

For what value of x will Neville be able to make an origami giraffe with the least amount of excess paper?

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