Algebra-Equations

From Exampleproblems

1 Linear
2 Quadratic
3 Solving Radical Equations
4 Solving for x
5 Factoring
6 Inequalities

Linear

solution Solve for $x$ : $x+4=10\,$

solution Solve for $x$ : $4x-17=2\,$

solution Solve for $x$ : $x^2-1=5\,$

solution Solve for $x$ : $23x-12=3x+10\,$

solution Solve for $x$ : $x-y+4x-y^2=12\,$

solution Solve the equations by eliminating in x $3x-5y=1,5x+2y=19\,$

solution Solve the equations by eliminating y in $3x-y=3,7x+2y=20\,$

solution Solve $23x-17y+11=0,31x+13y-57=0\,$

solution Find the solution such that u,v not equal to zero in $2u+v=\frac{7}{3}uv,u+3v=\frac{11}{3}uv\,$

solution Solve the following systems of linear equations in x and y $ax+by-a+b=0,bx-ay-a-b=0\,$

solution Solve $\frac{x}{a}+\frac{y}{b}-2=0,ax-by+b^2-a^2=0\,$

solution If twice the son's age in years is added to the age of his father,the sum is 90.If twice the father's age in years is added to the age of the son,the sum is 120.Find their ages.

solution The sum of a two-digit number and the number obtained by interchanging the digits of the number is 110.The digits of the number differ by 6.How many such numbers are there? Find all of them

solution If the numerator of a fraction is multiplied by 2 and its denominator is increased by 2,it becomes $\frac{6}{7}\,$ .If instead,we multiply the denominator by 2 and increase the numerator by 2,it reduces to $\frac{1}{2}\,$ .What is the fraction?

solution Robert purchased 5 chairs and 2 tables for $1625.Julie purchased 2 chairs and one table for $750.Find the cost per chair and table.

solution If we buy 2 air tickets from city A to city B and 3 tickets from city A to city C, we have to pay $1500. But 3 tickets from A to B and 5 tickets from A to C cost a total of $2000.What is the airfare from A to B and A to C?

solution The area of a rectangle gets reduced by 9 square units,if its length is reduced by 5 units and breadth is increased by 3 units.If we increase the length by 3 units and breadth by 2 units, then the area is increased by 67 square units. Find the length and breadth of the rectangle.

solution If in a rectangle the length is increased and breadth is reduced by 2 units each, the area is reduced by 28 square units.If the length is reduced by 1 unit and breadth is increased by 2 units, the area increases by 33 square units. Find the dimensions of the rectangle.

solution A person can row downstream 20kilo meters in 2 hours and upstream 4 km in 2 hours. Find the man's speed of rowing in still water and the speed of the current.

solution In a cyclic quadrilateral, $\angle A=(2x+4)^\circ,\angle B=(y+3)^\circ, \angle C=(2y+10)^\circ,\angle D=(4x-5y)^\circ\,$ . Find the four angles.

ALGEBRA BOOKS

Quadratic

solution Solve for $x$ : $\frac{1}{\sqrt{x^2-1}}=5\,$

solution Solve for $x$ : $x^2-x-6=0\,$

solution Solve for $x$ : $x^{-8}-17x^{-4}+16=0\,$

solution Solve for $x$ : $5x^2+6x+1=0\,$

solution Solve for $x$ : $x^2-1=0\,$

solution Solve for $x$ : $3x^2+1=13\,$

solution Solve $\frac{2x}{x-3}+\frac{1}{2x+3}+\frac{3x+9}{(x-3)(2x+3)}=0\,$

solution Write the discriminant of the quadratic equations $x^2+4x+1=0,3x^2+2x-1=0\,$

solution Determine the values of p in the following quadratic equations if they have real roots i). $px^2+4x+1=0\,$

ii). $5px^2-8x+2=0\,$

solution $2x^2-5x+2=0\,$

solution $y^2+7y+12=0\,$

solution $5x^2-12x+7=0\,$

solution $x^2-x-2=0\,$

solution $n^{-2}+3n^{-1}+2=0\,$

solution $x^{2/3}-x^{1/3}-6=0\,$

solution $6x^{2/3}-11x^{1/3}+4=0\,$

solution $x^{3/4}-x^{1/2}-x^{1/4}+1=0\,$

solution The sum of the squares of two consecutive positive integers is 545.Find the integers.

solution The length of the hall is 5meters more than its breadth.If the area of the floor of the hall is 84square meters,what are the length and breadth of the hall.

solution The hypotenuse of a triangle is 25cm.The difference between the lengths of the other two sides of the triangle is 5cm.Find the lengths of these sides.

solution A farmer wishes to grow a 100 square meters rectangular vegetable garden.Since he has with him only 30meter barbed wire,he fences three sides of the rectangular garden letting compound wall of his house act as the fourth side-fence.Find the dimensions of the garden.

solution If twice the area of a smaller square is subtracted from the area of a larger square,the result is 14 square cm.However,if the twice the area of the larger square is added to three times the area of the smaller square,the result is 203 square cm.Determine the sides of the two squares.

ALGEBRA BOOKS

Solving Radical Equations

solution Solve $\sqrt{x+2}=x-4\,$

solution Solve $\sqrt{3x+7}=3x+5\,$

solution Solve $\sqrt{x+1}+3=\sqrt{3x+4}\,$

Solving for x

solution Solve for $x$ : $\frac{4}{3}=\frac{7}{x}\,$

solution Solve for $x$ : $\frac{7}{x+2}=\frac{3}{2}\,$

solution Solve for $x$ : $\frac{1}{x+2}=\frac{5x}{9}\,$

solution Solve for $d_t\,$ : $\frac{1-d_t}{4d_t}=\frac{1}{2}\,$

solution Solve for $x$ : $2^{x+3}=4^{x+1}\,$

solution Solve for $x$ : $3^{x+3}=9^{x+1}\,$

solution Solve $2^{x+3}=4^{y-2}\,$ and $3^{x-2}=9^{3y-2x}\,$

solution If $a^{\frac{1}{3}}+b^{\frac{1}{3}}+c^{\frac{1}{3}}=0\,$ Show that $(a+b+c)^3=27abc\,$

solution Show that $\frac{1}{1+x^{a-b}+x^{a-c}}+\frac{1}{1+x^{b-c}+x^{b-a}}+\frac{1}{1+x^{c-a}+x^{c-b}}=1\,$

solution If $x=a^p,y=a^q\,$ and $x^q\cdot y^p=a^{\frac{2}{r}}\,$ .Show that $pqr=1\,$

solution If $a^x=b^y=c^z=d^w\,$ and $ab=cd\,$ .Show that $\frac{1}{x}+\frac{1}{y}=\frac{1}{w}+\frac{1}{z}\,$

solution If $a^x=b^y=c^z\,$ and $\frac{b}{a}=\frac{c}{b}\,$ ,show that $\frac{y}{x}=\frac{2z}{x+z}\,$

solution Simplify $[\frac{a^p}{a^q}]^{p+q}\cdot [\frac{a^q}{a^r}]^{q+r}\cdot [\frac{a^r}{a^p}]^{r+p}\,$

solution If $a=x+\sqrt{x^2+1}\,$ ,then show that $x=\frac{1}{2}(a-a^{-1})\,$

solution If $a^{x-1}=bc,b^{y-1}=ca,c^{z-1}=ab\,$ show that $xy+yz+zx=xyz\,$

ALGEBRA BOOKS

Factoring

Factoring polynomials of one variable using a form of the quadratic equation.

Situation

Suppose we have a polynomial

$a x 2 + b x + c$ where

a,b,c are constants

and we want to factor it

Method

calculate $Δ = b 2 - 4 a c$ if $Δ > 0$ calculate roots $x 1, x 2$

Conclusion

if roots have been found at the previous step we conclude that

$a x 2 + b x + c = a (x - x 1)(x - x 2)$

solution Factor $3x^3+24y^3\,$

solution Factor $8051\,$

solution Factor $x^2+8x+12\,$

solution Factor $x^2+6x-7\,$

solution Factor $x^2+2x-15\,$

solution Factor $x^2-4x+3\,$

solution Factorize $a^2-(b-c)^2\,$

solution Write the factors of $8a^3+36a^2+54a+27\,$

solution Factorize $x^4+x^2 y^2+y^4\,$

solution Factorize $125x^3-64\,$

solution Find the factors of $4a^2+4ab+b^2-c^2\,$

solution Factorize $8x^3-60x^2+150x-125\,$

solution Factorize $a^2 b^2+c^2 d^2-a^2 c^2-b^2 d^2\,$

solution Factorize $a^4-1\,$

Inequalities

solution $x^2 < 10 - 3x\,$

solution $\frac{x+4}{1-x}\le 0\,$

solution $x^2\le 4x\,$

solution $x^2+21 > 10x\,$

solution $\left | \frac{x+1}{x-1}\right \vert \le 2\,$