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Solving an Equation - Linear, Quadratic, Radical

#Algebra

TL;DR

Solving an equation means finding the value (or values) of the unknown that make both sides equal. The four-step universal process — simplify, isolate the variable, undo operations in reverse order, check — applies to every equation type.

What Does "Solving an Equation" Mean?

To solve an equation is to find the value(s) of the unknown variable that make the equation a true statement. The set of all such values is the solution set.

For example, the equation 2x+3=11 is true when x=4 (since 2⋅4+3=11) and false for any other value. So the solution set is {4}.

Some equations have:

• Exactly one solution — most linear equations.

• Multiple solutions — quadratics have up to two, cubics up to three.

• No solution — like ( x = -2 \sqrt{x} ) over the reals.

• Infinitely many solutions — identities like 2(x+3)=2x+6.

The Universal Process

Regardless of equation type:

1. Simplify each side — distribute, combine like terms, clear fractions or decimals.
2. Collect variable terms on one side, constants on the other.
3. Undo operations in reverse order to isolate the variable.
4. Check by substituting into the original equation.

How to Solve a Linear Equation

A linear equation has the variable to the first power: ( ax + b = c ).

Step-by-step:

1. Distribute any parentheses.
2. Combine like terms on each side.
3. Move variables to one side, constants to the other.
4. Divide by the coefficient of the variable.
5. Check.

Worked example. Solve ( 5(x + 2) - 3 = 17 ).

Step 1: Distribute → ( 5x + 10 - 3 = 17 ). Step 2: Combine → ( 5x + 7 = 17 ). Step 3: Subtract 7 → ( 5x = 10 ). Step 4: Divide by 5 → ( x = 2 ). Step 5: Check: ( 5(2 + 2) - 3 = 17 ) ✓.

How to Solve a Quadratic Equation

A quadratic equation has the variable to the second power as its highest term: ( ax^2 + bx + c = 0 ).

Three primary methods:

Method 1: Factoring

Rewrite the quadratic as a product of two linear factors. Use the zero-product property: if ( AB = 0 ), then ( A = 0 ) or ( B = 0 ).

Worked example. Solve ( x^2 - 7x + 12 = 0 ).

Factor: ( (x - 3)(x - 4) = 0 ). So ( x = 3 ) or ( x = 4 ).

Method 2: Quadratic Formula

For ( ax^2 + bx + c = 0 ):

( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )

Works for any quadratic — use when factoring isn't obvious.

Method 3: Completing the Square

Rewrite as a perfect square plus a constant. Mostly used for deriving the formula or in calculus.

How to Solve a Radical Equation

A radical equation has the variable inside a root: ( \sqrt{x + 5} = 3 ).

Process:

1. Isolate the radical.
2. Square both sides.
3. Solve the resulting equation.
4. Check for extraneous roots — squaring can introduce values that don't satisfy the original.

Worked example. Solve ( 2x + 7 = 5 \sqrt{2x + 7} = 5 ).

Step 1: Already isolated. Square both sides → ( 2x + 7 = 25 ). Step 2: Solve → ( x = 9 ). Step 3: Check: ( 2 \cdot 9 + 7 = 25 ) ✓.

Extraneous root example. Solve ( \sqrt{x} = -3 ).

Squaring gives ( x = 9 ). But ( \sqrt{9} = 3 ), not -3. So ( x = 9 ) is an extraneous root — and the original equation has no solution.

How to Solve a Rational Equation

A rational equation has the variable in a denominator: ( \frac{3}{x} = \frac{6}{x + 2} ).

Process:

1. Find the common denominator.
2. Multiply both sides by it to clear fractions.
3. Solve the resulting equation.
4. Check that no solution makes any original denominator zero.

Worked example. Solve ( \frac{3}{x} = \frac{6}{x + 2} ).

Cross-multiply: ( 3(x + 2) = 6x ) → ( 3x + 6 = 6x ) → ( 6 = 3x ) → ( x = 2 ). Check: at ( x = 2 ), neither ( x ) nor ( x + 2 ) is zero ✓.

How to Solve an Exponential Equation

An exponential equation has the variable in an exponent: ( 2^x = 32 ).

Process. Take the logarithm of both sides, then use the power rule.

Worked example. Solve ( 2^x = 32 ).

( \log_2(2^x) = \log_2(32) \implies x = 5 ).

Key Takeaways

• Solving an equation means finding the value(s) of the unknown that make both sides equal.
• Four universal steps: simplify each side, collect like terms, isolate the variable, check.
• Method depends on equation type: linear → legal moves; quadratic → factor/formula; radical → isolate and square; rational → clear denominators; exponential → take logs.
• Always check by substituting into the original equation, especially after squaring or multiplying by expressions containing the variable.
• Some equations have no solution (inconsistent), some have infinitely many (identities), some have multiple (polynomial equations).

Frequently Asked Questions

What is the first step in solving any equation?

Simplify each side separately — distribute, combine like terms, clear fractions. Don't move anything between sides until each side is in its simplest form.

How do I know if an equation has no solution?

If the equation simplifies to a false statement (like 0 = 1), it has no solution. If it simplifies to a true statement (like 0 = 0), it has infinitely many solutions.

When should I use the quadratic formula vs factoring?

Try factoring first — it's faster when the factors are integers. Use the formula when factoring isn't obvious or when the roots aren't rational.

What's an extraneous root?

A value that satisfies the transformed equation but not the original. Introduced when you square both sides or multiply by an expression. Always check.

Can equations have complex solutions?

Yes — a quadratic with negative discriminant has complex (non-real) solutions. The fundamental theorem of algebra guarantees every polynomial of degree n has n complex roots.