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Algebra Worksheet
USEFUL FORMULAS:
Arithmetic formulas
(a−b)(a+b) = a^{2}−b^{2},
(a+b)^{2} = a^{2}+2ab+b^{2},
(a−b)^{2} = a^{2}−2ab+b^{2}.
Rules concerning logarithms
Rules about solving equations
• Whenever you divide by some number make sure it is not 0!
• If a fraction is 0 then this means its numerator is 0, i.e.
• If a product is equal to 0 than one of its parts is zero , so check
all of them, i.e.
f(x) · g(x) = 0 => solve f(x) = 0 and g(x) = 0.
• Know the quadratic formula and remember that if x^{2}+bx+c
= 0
has roots
and
then x^{2} + bx + c = (x −
) · (x −β
).
1. Decide if the following are TRUE or FALSE.
2. Solve for t: (t − 5)(t + 1) = 11
3. Solve for θ:
4. Solve for y in terms of x : x^{2} + 3y = −7xy + 11.
5. Does the following equation determine y as a function
of x: y^{2} −
x^{3} = αx.
6. Solve for T and simplify as much as possible:
7. The following sequence of exercises is meant to review
basic identities
that you should know.
Start with (x + 2)^{2} and do the multiplication, you should get x^{2} +
4x + 4.
Now do the same for (x + 3)^{2}, (2 + x)^{2} and (x + y)^{2}.
At this point you should see that there is an actual formula that one
can use in for each of these. The formula is
(A + B)^{2} = A^{2} + 2AB + B^{2}.
Next, start with (x − 2)^{2} and do the multiplication, you
should get
x^{2} − 4x + 4.
Now do the same for (x − 3)^{2}, (2 − x)^{2} and (x − y)^{2}.
Again you should see that there is a convenient formula
(A − B)^{2} = A^{2} − 2AB + B^{2}.
The two formulas above are the basic tools for completing
the square.
There is one more formula that is important. Here is a way to get it.
Multiply (x − 1)(x + 1), what do you get?
Now do the same for (x+2)(x−2), (x−7)(x+7), and (x−y)(x+y).
Do you see a formula?
You always have
(A − B)(A + B) = A^{2} − B^{2}.
Now that you know the basic formulas let’s do some
exercises with
them. The next exercise aims to do exactly this.
8. Simplify
9. Write as a single log
log z ^{2} + log 10 + log(y − x);
7 log y − 12 logα − 11 log e.
10. Find or simplify the following expressions :
(A)
(B) (x − 7)(x + 7) + 49 =?
(C) −(x − 5)^{2} =?
(D) (3y + 1)(1 − 3y) =?
(E) (y^{3} − 3)(y^{3} + 3) =?
11. Factor the following expressions
(A) y(y − 3) + 15(3 − y)
(B) 7x^{2} + 14x − 21
(C)
(D) x^{2} − x − 1
(E) (x − 1)(x + 1) + x^{2} − 2x + 1.
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