In problems asking for numerical or symbolic answer, you have to prove (same as
as ‘justify’) your method . Sometimes we remind about this in the problem’s
we do not. Often you can just refer to the main facts proved in this handout.
The statement of the
form ‘If A, then B’ means the same as ‘Given A, prove B’. Proof of the statement
of the form ‘A if
and only if B’ usually amounts to two proofs : ‘If A, then B’ and ‘If B, then A’.
1. For each of the following quadratic polynomials (a) complete the square; (b)
find the minimum or
the maximum values of the corresponding functions and the value of x for which
it is attained;
(c) find the roots (i.e. solve the corresponding quadratic equation); (d) sketch
(ii) x2 + 6x + 9
(iii) x2 + 6x
(iv) −2x2 − 5
(v) −2x2 + 5
(vi) x2 − 2x + 3
(vii) 3x2 + 4x − 1
(viii) −2x2 + 4x − 1
2. Prove that a quadratic function f(x) = ax2 + bx + c, with a > 0 (a < 0),
(with the domain R -
the set of all real numbers ) always attains the minimum (maximum) value exactly
for one value of
x, and does not attain maximum (minimum) value.
What is this value of x?
3. (i) Among all pairs of two real numbers whose sum is 10, find the pair with
the greatest product.
Justify. What is the maximum value of the product ? What can be said about the
of the product?
(ii) Among all rectangles with perimeter 20 find the one of the greatest area.
What is the maximum
Generalize, by replacing 20 with an arbitrary positive number p.
4. Find all pairs of real numbers (x, y) such that x2 + y2 + 6x = 10y − 34.
5. Let x1 and x2 be the roots of the quadratic polynomial f(x) = 5x2 − 20x − 1.
(i) 2(x1 + x2)
(iv) 1/x1 + 1/x2
6. Factor the following quadratic polynomials into the product of the first
degree polynomials with
real coefficients and with the coefficients at x equal to 1, or show that no
such factorization exists.
7. Find the coordinates of the points of intersections (or show that they do not
exist) of the line
y = 2x + 1 and the following parabolas. You do not have to draw graphs to solve
but it is instructive to draw them after the solution is obtained in order to
see what happens
(iv) (2, 5), (−1, 7), and has its vertex at (1, 4).
10. For which values of r, the equation x2 + 4rx + (5 + 4r) = 0 has
(i) two distinct real roots?
(ii) two equal real roots?
11. For which values of r, the equation x2 + (4 + 2r)x + (5 + 4r) = 0 has
(i) equal roots?
(ii) opposite roots? (i.e., with equal absolute values, but
different signs .)
12. Prove that for every three non-collinear points (x1, y1), (x2, y2), (x3,
y3), (i.e., the points are
not on a line) and with pairwise distinct x-coordinates, there exists exactly
one parabola passing
Will the conclusion hold if we allow the three points to be collinear? Will the
conclusion hold if we
allow two of the points lie on a vertical line?
13. For which values of a, the line y = 3x + a and the parabola y = x2 + x have
(i) two common
points? (ii) one common point ? (iii) no common points?
14. Prove that the graph of the parabola y = f(x) = ax2 + bx + c is symmetric
with respect to the
vertical line x = −b/2a.
The line x = −b/2a is called the axis of the parabola.
(Hint: First realize that it amounts showing that the points of the graph whose
symmetric with respect to the number −b/2a have equal y-coordinates.
Symbolically this means that
for every number h, f(−b/2a − h) = f(−b/2a + h). Therefore we have to check the
15. Prove that for all real numbers x, y, the polynomial x2 +xy +y2
≥ 0 and that
it takes value zero
if and only if x = y = 0.
16. (i) If 2a + 3b = 12, what is the greatest value of ab? Justify.
(ii) If ab = 12, what is the smallest value of a2 + b2? Justify.
17. Find all value of a for which the polynomials x2+ax+1 and x2+x+a have at
least one common
18. Prove that for all real numbers x, y, z, x2 + y2 + z2
≥ xy + yz + zx and the
equality sign is
attained if and only if x = y = z.
19. Prove that for all real numbers x, y, z, x + y + z = 1 implies that x2
+ z2 ≥ 1/3, and the
equality sign is attained if and only if x = y = z.
20. Prove that for all real numbers x, y, z, x + y + z ≥ 0 implies x3 + y3 + z3
≥ 3xyz. Then show
that x3 + y3 + z3 = 3xyz if and only if x = y = z or x + y + z = 0.
21. Among all rectangles with area a > 0, find the one of the smallest perimeter
(i.e., determine the
lengths of its sides).
22. A stone moves in such a way that its height after t seconds of its motion is
given by the formula
h(t) = −16t2 +96t+256 feet. How high will the stone go? At what moment(s) of
time will it be
144 feet high? When does it hit the ground?
Start solving your Algebra Problems
in next 5 minutes!
Download (and optional CD)
Click to Buy Now:
2Checkout.com is an authorized reseller
of goods provided by Sofmath
Attention: We are
currently running a special promotional offer
for Algebra-Answer.com visitors -- if you order
Algebra Helper by midnight of
you will pay only $39.99
instead of our regular price of $74.99 -- this is $35 in
savings ! In order to take advantage of this
offer, you need to order by clicking on one of
the buttons on the left, not through our regular
If you order now you will also receive 30 minute live session from tutor.com for a 1$!
You Will Learn Algebra Better - Guaranteed!
Just take a look how incredibly simple Algebra Helper is:
: Enter your homework problem in an easy WYSIWYG (What you see is what you get) algebra editor:
Step 2 :
Let Algebra Helper solve it:
Step 3 : Ask for an explanation for the steps you don't understand:
Algebra Helper can solve problems in all the following areas:
simplification of algebraic expressions (operations
with polynomials (simplifying, degree, synthetic division...), exponential expressions, fractions and roots
(radicals), absolute values)