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Equations

This page is part of the section “USE basic level. Equations”. Here are solutions to typical equations: linear, quadratic, rational and exponential. All problems are written independently, with neat answers.

Problems with solutions

Problem 1. Linear equation

Solve the equation:

4x−9=2x+3

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

4x−2x=3+9⇒2x=12⇒x=6

Answer: x=6.

Problem 2. Quadratic equation

Solve the equation:

x2−7x+10=0

By Vieta’s theorem the sum of the roots is 7, the product is 10:

x1+x2=7,x1·x2=10

We look for two numbers: 2 and 5.

Answer: x1=2, x2=5.

Problem 3. Rational equation

Solve the equation:

x+34=2x−13

Multiply both sides by 12 (the least common multiple of 4 and 3) to eliminate the fractions:

3(x+3)=4(2x−1) 3x+9=8x−4⇒−5x=−13⇒x=2.6

Move the terms with x to the left, the numbers to the right, and find x:

Answer: x=2.6.

Problem 4. Exponential equation

Solve the equation:

3x−2=81

Represent 81 as a power of three: 81 = 3⁴. We get:

81=34⇒3x−2=34

Equate the exponents:

x−2=4⇒x=6

Answer: x=6.

Frequently asked questions

How to solve linear equations from the USE basic level?

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.

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. For a complete equation, the discriminant is easier.

How to eliminate fractions in an equation?

Multiply both sides of the equation by the least common multiple of the denominators. Then the fractions disappear and the equation reduces to a linear one without fractions.

How to solve exponential equations from the USE basic level?

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

See also