# Quadratic Equations: The Complete Guide to Solving and Understanding Quadratic equations are among the most important and frequently encountered equations in mathematics. From calculating projectile trajectories to optimizing business profits, quadratic equations appear throughout science, engineering, and everyday problem-solving. This comprehensive guide will teach you everything you need to know about understanding, solving, and applying quadratic equations. ## What is a Quadratic Equation? A quadratic equation is a polynomial equation of degree 2, meaning the highest power of the variable is 2. ### Standard Form The standard form of a quadratic equation is: **ax² + bx + c = 0** Where: - **a, b, c** are constants (real numbers) - **a ≠ 0** (if a = 0, it becomes a linear equation) - **x** is the variable (unknown) ### Examples - x² - 5x + 6 = 0 (a=1, b=-5, c=6) - 2x² + 3x - 2 = 0 (a=2, b=3, c=-2) - x² - 16 = 0 (a=1, b=0, c=-16) - 3x² + 12x = 0 (a=3, b=12, c=0) ### Other Forms **Vertex Form:** y = a(x - h)² + k - (h, k) is the vertex of the parabola **Factored Form:** y = a(x - r₁)(x - r₂) - r₁ and r₂ are the roots (solutions) ## Understanding Solutions (Roots) The solutions to a quadratic equation are called **roots**, **zeros**, or **x-intercepts**. A quadratic equation can have: - **Two distinct real roots** (discriminant > 0) - **One repeated real root** (discriminant = 0) - **Two complex roots** (discriminant < 0) The **discriminant** is: **Δ = b² - 4ac** ## Method 1: Solving by Factoring Factoring is often the fastest method when the quadratic can be easily factored. ### The Zero Product Property If AB = 0, then either A = 0 or B = 0 (or both). This means: if (x - r₁)(x - r₂) = 0, then x = r₁ or x = r₂ ### Simple Factoring (a = 1) **Example 1:** Solve x² + 5x + 6 = 0 ``` Step 1: Factor the left side Find two numbers that multiply to 6 and add to 5: 2 and 3 (x + 2)(x + 3) = 0 Step 2: Apply zero product property x + 2 = 0 OR x + 3 = 0 x = -2 OR x = -3 Solutions: x = -2, x = -3 ``` **Example 2:** Solve x² - 7x + 12 = 0 ``` Step 1: Factor Numbers: -3 and -4 (multiply to 12, add to -7) (x - 3)(x - 4) = 0 Step 2: Solve x = 3 or x = 4 Solutions: x = 3, x = 4 ``` ### Factoring with a ≠ 1 **Example 3:** Solve 2x² + 7x + 3 = 0 ``` Step 1: Factor using AC method ac = 2 × 3 = 6 Find numbers that multiply to 6 and add to 7: 1 and 6 Step 2: Rewrite and group 2x² + x + 6x + 3 = 0 x(2x + 1) + 3(2x + 1) = 0 (2x + 1)(x + 3) = 0 Step 3: Solve 2x + 1 = 0 OR x + 3 = 0 x = -1/2 OR x = -3 Solutions: x = -1/2, x = -3 ``` ### Special Cases **Difference of Squares:** x² - 9 = 0 (x + 3)(x - 3) = 0 **x = 3 or x = -3** **Perfect Square:** x² + 6x + 9 = 0 (x + 3)² = 0 **x = -3** (double root) Use our [quadratic factorization calculator](/tools/factor-quadratic) to factor and solve quadratic equations instantly. ## Method 2: The Quadratic Formula The quadratic formula works for **any** quadratic equation, even when factoring is difficult or impossible. ### The Formula For ax² + bx + c = 0: **x = [-b ± √(b² - 4ac)] / (2a)** ### Steps to Use the Formula 1. Write the equation in standard form: ax² + bx + c = 0 2. Identify a, b, and c 3. Calculate the discriminant: b² - 4ac 4. Substitute into the formula 5. Simplify (two solutions from ± symbol) ### Example 1: Two Distinct Roots Solve: x² - 3x - 10 = 0 ``` Step 1: Identify coefficients a = 1, b = -3, c = -10 Step 2: Calculate discriminant Δ = (-3)² - 4(1)(-10) = 9 + 40 = 49 Step 3: Apply formula x = [3 ± √49] / 2 x = [3 ± 7] / 2 Step 4: Solve x = (3 + 7)/2 = 5 OR x = (3 - 7)/2 = -2 Solutions: x = 5, x = -2 ``` ### Example 2: One Repeated Root Solve: x² - 6x + 9 = 0 ``` a = 1, b = -6, c = 9 Δ = (-6)² - 4(1)(9) = 36 - 36 = 0 x = [6 ± √0] / 2 = 6/2 = 3 Solution: x = 3 (double root) ``` ### Example 3: No Real Roots Solve: x² + 2x + 5 = 0 ``` a = 1, b = 2, c = 5 Δ = 2² - 4(1)(5) = 4 - 20 = -16 < 0 Since discriminant is negative, no real solutions exist. Complex solutions: x = [-2 ± √(-16)] / 2 = -1 ± 2i ``` ### Example 4: Non-Integer Coefficients Solve: 2x² + 5x - 1 = 0 ``` a = 2, b = 5, c = -1 Δ = 5² - 4(2)(-1) = 25 + 8 = 33 x = [-5 ± √33] / 4 Solutions: x = (-5 + √33)/4 ≈ 0.186 x = (-5 - √33)/4 ≈ -2.686 ``` ## Method 3: Completing the Square This method transforms a quadratic into perfect square form, useful for deriving the quadratic formula and graphing. ### The Process 1. Move the constant term to the right side 2. If a ≠ 1, divide everything by a 3. Take half of the b coefficient, square it, and add to both sides 4. Factor the left side as a perfect square 5. Take the square root of both sides 6. Solve for x ### Example 1: Basic Completing the Square Solve: x² + 8x + 7 = 0 ``` Step 1: Move constant x² + 8x = -7 Step 2: Complete the square Take half of 8: 8/2 = 4 Square it: 4² = 16 Add to both sides: x² + 8x + 16 = -7 + 16 Step 3: Factor left side (x + 4)² = 9 Step 4: Take square root x + 4 = ±3 Step 5: Solve x = -4 + 3 = -1 OR x = -4 - 3 = -7 Solutions: x = -1, x = -7 ``` ### Example 2: With Leading Coefficient ≠ 1 Solve: 2x² - 12x + 10 = 0 ``` Step 1: Divide by a (2) x² - 6x + 5 = 0 Step 2: Move constant x² - 6x = -5 Step 3: Complete the square (-6/2)² = 9 x² - 6x + 9 = -5 + 9 (x - 3)² = 4 Step 4: Solve x - 3 = ±2 x = 3 + 2 = 5 OR x = 3 - 2 = 1 Solutions: x = 5, x = 1 ``` ### Converting to Vertex Form Completing the square converts y = ax² + bx + c to vertex form: y = a(x - h)² + k **Example:** Convert y = x² - 4x + 1 to vertex form ``` y = x² - 4x + 1 y = (x² - 4x + 4) + 1 - 4 y = (x - 2)² - 3 Vertex form: y = (x - 2)² - 3 Vertex: (2, -3) ``` ## Method 4: Graphing Quadratic equations graph as parabolas. The solutions are where the parabola crosses the x-axis. ### Key Features of a Parabola **Vertex:** The highest or lowest point - If a > 0: parabola opens upward (vertex is minimum) - If a < 0: parabola opens downward (vertex is maximum) **Axis of Symmetry:** x = -b/(2a) **Y-intercept:** (0, c) **X-intercepts (roots):** Solutions to ax² + bx + c = 0 ### Finding the Vertex **Method 1:** Use h = -b/(2a), then find k = f(h) **Method 2:** Complete the square to get vertex form y = a(x - h)² + k **Example:** Find the vertex of y = 2x² - 8x + 3 ``` Method 1: h = -(-8)/(2·2) = 8/4 = 2 k = 2(2)² - 8(2) + 3 = 8 - 16 + 3 = -5 Vertex: (2, -5) ``` ### Analyzing Solutions from the Graph - **Graph crosses x-axis twice:** Two real roots - **Graph touches x-axis once:** One repeated root - **Graph doesn't touch x-axis:** No real roots ## Choosing the Best Method | Situation | Best Method | |-----------|-------------| | Easy to factor | Factoring | | a = 1, small integers | Factoring | | Not factorable | Quadratic Formula | | Need exact irrational answers | Quadratic Formula | | Finding vertex | Completing the Square | | Deriving formulas | Completing the Square | | Visual understanding | Graphing | | c = 0 | Factor out x first | | b = 0 | Take square root directly | ## Real-World Applications ### 1. Projectile Motion **Problem:** A ball is thrown upward with initial velocity 20 m/s from a height of 2 m. When does it hit the ground? **Equation:** h(t) = -5t² + 20t + 2 = 0 (where h is height in meters, t is time in seconds) ``` Using quadratic formula: a = -5, b = 20, c = 2 t = [-20 ± √(400 + 40)] / (-10) t = [-20 ± √440] / (-10) t = [-20 ± 20.98] / (-10) t = 0.098 s (going up, ignoring) t = 4.098 s (hitting ground) Answer: Approximately 4.1 seconds ``` ### 2. Area Problems **Problem:** A rectangular garden has length 4 meters more than its width. If the area is 60 square meters, find the dimensions. ``` Let width = x Then length = x + 4 Area: x(x + 4) = 60 x² + 4x = 60 x² + 4x - 60 = 0 Factoring: (x + 10)(x - 6) = 0 x = -10 (reject, negative) or x = 6 Width: 6 meters Length: 10 meters ``` ### 3. Business Optimization **Problem:** A company's profit P (in thousands) is given by P(x) = -2x² + 40x - 150, where x is units produced (in thousands). Find maximum profit. ``` Vertex represents maximum (a < 0) h = -40/(2·(-2)) = 40/4 = 10 k = -2(10)² + 40(10) - 150 = -200 + 400 - 150 = 50 Maximum profit: $50,000 at 10,000 units ``` ### 4. Engineering and Design **Problem:** A bridge arch follows the equation y = -0.01x² + 2x, where y is height in meters. Find the span of the arch (where it meets the ground). ``` Set y = 0: -0.01x² + 2x = 0 x(-0.01x + 2) = 0 x = 0 (one end) -0.01x + 2 = 0 x = 200 (other end) Span: 200 meters ``` ### 5. Physics: Acceleration **Problem:** A car's position is given by s(t) = 3t² + 2t + 1 meters. When is it at position 50 meters? ``` 3t² + 2t + 1 = 50 3t² + 2t - 49 = 0 Using quadratic formula: t = [-2 ± √(4 + 588)] / 6 t = [-2 ± √592] / 6 t = [-2 ± 24.33] / 6 t = 3.72 seconds (positive solution) ``` ## Sum and Product of Roots For ax² + bx + c = 0 with roots r₁ and r₂: **Sum of roots:** r₁ + r₂ = -b/a **Product of roots:** r₁ · r₂ = c/a ### Application **Example:** Without solving, find the sum and product of roots of 3x² - 12x + 5 = 0 ``` Sum: -(-12)/3 = 12/3 = 4 Product: 5/3 ``` This is useful for: - Checking solutions - Creating equations from given roots - Analyzing equations without solving ## Creating Equations from Roots If you know the roots, you can construct the equation. **Formula:** (x - r₁)(x - r₂) = 0 **Example:** Create a quadratic with roots 3 and -5 ``` (x - 3)(x - (-5)) = 0 (x - 3)(x + 5) = 0 x² + 5x - 3x - 15 = 0 x² + 2x - 15 = 0 ``` ## Common Mistakes to Avoid ### Mistake 1: Forgetting ± in Square Root **Wrong:** x² = 9, so x = 3 **Right:** x² = 9, so x = ±3 (x = 3 or x = -3) ### Mistake 2: Sign Errors in Formula **Wrong:** For x² - 5x + 6 = 0, using b = 5 **Right:** b = -5 (include the sign!) ### Mistake 3: Dividing by Variable **Wrong:** x² = 3x, so x = 3 (dividing both sides by x) **Right:** x² - 3x = 0, so x(x - 3) = 0, giving x = 0 or x = 3 ### Mistake 4: Incomplete Factoring **Wrong:** 2x² + 4x = 0 → (x + 2)(2x) = 0 **Right:** 2x² + 4x = 0 → 2x(x + 2) = 0 ### Mistake 5: Arithmetic Errors Always double-check: - Discriminant calculation - Signs when substituting - Simplifying radicals ## Practice Problems ### Beginner: 1. Solve: x² + 7x + 12 = 0 2. Solve: x² - 16 = 0 3. Solve: x² + 4x = 0 ### Intermediate: 4. Solve: 2x² - 5x - 3 = 0 5. Solve: x² - 4x + 4 = 0 6. Solve: x² + 6x + 2 = 0 ### Advanced: 7. Find the vertex of y = -3x² + 12x - 5 8. A rectangle's length is 3 more than twice its width, and area is 44. Find dimensions. 9. Solve: (x + 1)² = 5 ### Answers: 1. x = -3, x = -4 2. x = 4, x = -4 3. x = 0, x = -4 4. x = 3, x = -1/2 5. x = 2 (double root) 6. x = -3 ± √7 7. Vertex: (2, 7) 8. Width 4, Length 11 9. x = -1 ± √5 ## Advanced Topics ### Complex Roots When Δ < 0, solutions involve imaginary unit i where i² = -1. **Example:** x² + 2x + 5 = 0 ``` x = [-2 ± √(4-20)] / 2 x = [-2 ± √(-16)] / 2 x = [-2 ± 4i] / 2 x = -1 ± 2i Complex roots: -1 + 2i and -1 - 2i (complex conjugates) ``` ### Quadratic Inequalities Solve: x² - 5x + 6 < 0 ``` Step 1: Factor (x - 2)(x - 3) < 0 Step 2: Find critical points x = 2, x = 3 Step 3: Test intervals x < 2: (+)(+) = + (false) 2 < x < 3: (-)(+) = - (true) x > 3: (+)(+) = + (false) Solution: 2 < x < 3 ``` ## Conclusion Quadratic equations are a cornerstone of algebra with applications across mathematics, science, and engineering. Mastering the various solution methods—factoring, quadratic formula, completing the square, and graphing—gives you flexibility to approach problems efficiently. Key takeaways: - Choose the method that best fits the problem - Always check your solutions by substituting back - Understand the discriminant to predict the nature of roots - Recognize real-world applications to motivate learning - Practice regularly with diverse problems Whether you're solving for projectile motion, optimizing business scenarios, or analyzing parabolic structures, the skills you've learned here will serve you well. Use our [quadratic calculator](/tools/factor-quadratic) to practice and verify your solutions as you build confidence in working with quadratic equations. Remember: every method has its place, and true mastery comes from knowing which tool to use when. Keep practicing, and quadratic equations will become second nature!

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