Book A Free Math Class # Solving an Equation - Linear, Quadratic, Radical [#Algebra](/content/tag/algebra/index.html) 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.