How to Solve Quadratic Equations - Complete Guide
Learn how to solve quadratic equations using factoring, the quadratic formula, and completing the square. Step-by-step methods with worked examples.
What Is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree two, meaning the highest power of the variable is 2. The standard form is ax² + bx + c = 0, where a, b, and c are constants and a is not equal to zero. Quadratic equations arise naturally in physics (projectile motion), engineering (structural design), economics (profit optimization), and pure mathematics. Every quadratic equation has exactly two solutions (counting multiplicity and including complex numbers), although those solutions may be real or complex, distinct or repeated.
Solving by Factoring
Factoring is the fastest method when it works. The goal is to rewrite ax² + bx + c as a product of two binomials, such as (px + q)(rx + s) = 0. Once factored, set each factor equal to zero and solve for x. For example, x² - 5x + 6 = 0 factors as (x - 2)(x - 3) = 0, giving solutions x = 2 and x = 3. To factor, look for two numbers that multiply to give a*c and add to give b. This method is efficient for simple integer-coefficient equations but becomes impractical when the roots are irrational or complex. Always check your factoring by expanding the product back out.
The Quadratic Formula
The quadratic formula works for every quadratic equation, regardless of whether it can be factored neatly. Given ax² + bx + c = 0, the solutions are x = (-b plus or minus the square root of (b² - 4ac)) / (2a). This formula is derived by completing the square on the general equation. For example, to solve 2x² + 3x - 5 = 0, substitute a = 2, b = 3, c = -5: the discriminant is 9 + 40 = 49, so x = (-3 plus or minus 7) / 4, giving x = 1 and x = -2.5. Memorizing this formula is one of the most valuable investments you can make in algebra.
The Discriminant
The discriminant, b² - 4ac, tells you the nature of the solutions before you even solve the equation. If the discriminant is positive, there are two distinct real solutions, and the parabola crosses the x-axis at two points. If the discriminant is zero, there is exactly one repeated real solution (a "double root"), and the parabola just touches the x-axis at its vertex. If the discriminant is negative, there are no real solutions; the two solutions are complex conjugates, and the parabola does not intersect the x-axis at all. Checking the discriminant first can save time and guide your solving strategy.
Completing the Square
Completing the square transforms ax² + bx + c = 0 into a perfect square trinomial, making it easy to solve by taking square roots. Start by dividing through by a (if a is not 1), then move c to the other side of the equation. Take half of the coefficient of x, square it, and add it to both sides. This creates a perfect square on the left: (x + b/(2a))² = (b² - 4ac)/(4a²). Take the square root of both sides and solve for x. For example, x² + 6x + 2 = 0 becomes (x + 3)² = 7, so x = -3 plus or minus the square root of 7. This method is also used to convert quadratic functions to vertex form for graphing.
Solving by Graphing
Graphing provides a visual approach to finding solutions. The solutions to ax² + bx + c = 0 are the x-intercepts of the parabola y = ax² + bx + c. Plot the function, and the points where it crosses the x-axis are the real solutions. If the parabola does not cross the x-axis, the equation has no real solutions. While graphing alone may not give exact answers (especially for irrational roots), it provides excellent intuition about the number and approximate location of solutions. Modern graphing calculators and online tools can pinpoint intersections to many decimal places.
Special Forms and Shortcuts
Certain quadratic equations have quick-solve shortcuts. If there is no linear term (b = 0), the equation ax² + c = 0 simplifies to x² = -c/a, which you solve by taking square roots. If there is no constant term (c = 0), factor out x: ax² + bx = x(ax + b) = 0, giving x = 0 and x = -b/a immediately. Perfect square trinomials like x² + 10x + 25 = (x + 5)² = 0 have a single repeated root. Difference-of-squares patterns like x² - 9 = (x + 3)(x - 3) = 0 factor instantly. Recognizing these special forms saves significant time.
Applications of Quadratic Equations
Quadratic equations model many real-world scenarios. In physics, the height of a projectile follows h(t) = -16t² + v₀t + h₀ (in feet), so finding when the projectile hits the ground means solving a quadratic. In business, profit functions are often quadratic, and finding break-even points requires solving the equation where profit equals zero. In geometry, problems involving area frequently lead to quadratics, such as finding the dimensions of a rectangle with a given area and perimeter. Understanding how to solve these equations is a fundamental skill that connects abstract algebra to practical problem solving.
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