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Algebra

This page is part of the section “BSE. Algebra”. Here are solutions to typical algebra problems: linear and quadratic equations, simplifying expressions, exponential equations. All problems are written independently, with neat integer answers.

Problems with solutions

Problem 1. Linear equation

Solve the equation:

2x−5=3x+1

Move the terms with x to the left side and the numbers to the right side:

2x−3x=1+5⇒−x=6⇒x=−6

Answer: x=−6.

Problem 2. Quadratic equation

Solve the equation:

x2−6x+8=0

By Vieta’s theorem the sum of the roots is 6, the product is 8:

x1+x2=6,x1·x2=8

We look for two numbers: 2 and 4.

Answer: x1=2, x2=4.

Problem 3. Simplifying an expression

Simplify the expression:

(a+3)2−a(a+6)

Expand the square of the sum using (a + b)² = a² + 2ab + b² and expand the brackets in the second term:

(a2+6a+9)−(a2+6a)

Combine like terms:

a2+6a+9−a2−6a=9

Answer: 9.

Problem 4. Exponential equation

Solve the equation:

5x=125

Represent 125 as a power of five: 125 = 5³. We get:

125=53⇒5x=53⇒x=3

Answer: x=3.

Frequently asked questions

How to solve linear equations on the BSE?

Expand the brackets, move the terms with x to one side and the numbers to the other, combine like terms and find x. When a term is moved across the equals sign, its sign changes to the opposite.

How to simplify expressions with abbreviated multiplication formulas?

Expand the brackets using the formulas (a ± b)² = a² ± 2ab + b², a² − b² = (a − b)(a + b), then combine like terms. The higher powers often cancel out, and the answer is a number.

When is it convenient to use Vieta’s theorem?

When the equation is reduced (a = 1) and the roots are integers. Then the sum of the roots is −b and the product is c, and the roots are found mentally.

How to solve exponential equations on the BSE?

Reduce both sides to a common base. If a^u = a^v, then u = v. On the BSE, a common base (2, 3, 5, 10) can almost always be found.

See also