Solving Quadratic Equations

Methods of Solving Quadratic Equations This guide covers the main algebraic techniques for solving quadratic equations of the form $ax^2 + bx + c = 0$ where $a \neq 0$. Each method has its strengths, and understanding when to use each one is an important skill. Factoring by Inspection What it does: Factoring rewrites a quadratic expression as a product of two linear factors, which you can then solve using the Zero Factor Property. The idea: Any quadratic $ax^2 + bx + c$ can be written as $(px + q)(rx + s)$ where the numbers satisfy three conditions: $pr = a$ (the leading coefficient) $qs = c$ (the constant term) $ps + qr = b$ (the middle coefficient) When you have the factored form $(px + q)(rx + s) = 0$, the Zero Factor Property tells us that the product equals zero if and only if at least one factor equals zero. This gives us two equations to solve: $px + q = 0$ or $rx + s = 0$. Example with a monic quadratic: For a monic quadratic $x^2 + bx + c$ (where the coefficient of $x^2$ is 1), we're looking for factors $(x + q)(x + s)$ where: $q + s = b$ (they add to the middle coefficient) $qs = c$ (they multiply to the constant term) This relationship is known as Vieta's rule and makes monic quadratics easier to factor. For instance, $x^2 + 5x + 6$ factors as $(x + 2)(x + 3)$ because $2 + 3 = 5$ and $2 \times 3 = 6$. When to use it: Factoring is fastest when the quadratic factors nicely into integers. However, not all quadratics can be factored using rational numbers, so you'll need other methods in those cases. Completing the Square What it does: This method transforms a quadratic equation into a perfect square trinomial, which can be solved by taking square roots. Why it works: A perfect square trinomial like $(x + h)^2$ expands to $x^2 + 2hx + h^2$. By manipulating your original equation, you can create this form and then use square roots to solve. The step-by-step process: Divide by $a$: Start with $ax^2 + bx + c = 0$ and divide every term by $a$: $$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$$ Move the constant: Subtract $\frac{c}{a}$ from both sides: $$x^2 + \frac{b}{a}x = -\frac{c}{a}$$ Complete the square: Add $\left(\frac{b}{2a}\right)^2$ to both sides (this is the key step—you're adding half the coefficient of $x$, squared): $$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$ The left side is now a perfect square: $$\left(x + \frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$ Take square roots: Apply the square root to both sides, remembering the $\pm$ sign: $$x + \frac{b}{2a} = \pm\sqrt{-\frac{c}{a} + \left(\frac{b}{2a}\right)^2}$$ Solve for $x$: Subtract $\frac{b}{2a}$ from both sides to isolate $x$. Example: For $x^2 + 6x + 5 = 0$: Move the constant: $x^2 + 6x = -5$ Complete the square: $x^2 + 6x + 9 = -5 + 9 = 4$ Factor: $(x + 3)^2 = 4$ Take square roots: $x + 3 = \pm 2$ Solutions: $x = -3 + 2 = -1$ or $x = -3 - 2 = -5$ When to use it: Completing the square is useful when you need to understand the structure of the equation, such as finding the vertex of a parabola. It's also the method that leads directly to deriving the quadratic formula. The Quadratic Formula What it does: This formula provides a direct way to find the solutions of any quadratic equation. The formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This formula comes from applying the completing the square method to the general quadratic $ax^2 + bx + c = 0$. The key advantage is that you don't have to do any algebra yourself—just plug in your values for $a$, $b$, and $c$. The discriminant: The expression $b^2 - 4ac$ under the square root is called the discriminant, often denoted as $D$. This number is crucial because it tells you what type of solutions you'll get (we'll discuss this more in the next section). Example: For $2x^2 + 3x - 2 = 0$, we have $a = 2$, $b = 3$, and $c = -2$: $$x = \frac{-3 \pm \sqrt{9 - 4(2)(-2)}}{2(2)} = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4}$$ This gives us $x = \frac{2}{4} = \frac{1}{2}$ or $x = \frac{-8}{4} = -2$. When to use it: The quadratic formula always works for any quadratic equation. Use it when factoring isn't obvious or when you need a quick answer without worrying about the structure of the problem. The Reduced Quadratic Equation What it is: Sometimes it's convenient to divide the standard form $ax^2 + bx + c = 0$ by $a$ (remember, $a \neq 0$) to get the reduced monic equation: $$x^2 + px + q = 0$$ where $p = \frac{b}{a}$ and $q = \frac{c}{a}$. The quadratic formula for reduced form: For this simplified equation, the quadratic formula becomes: $$x = \frac{-p \pm \sqrt{p^2 - 4q}}{2}$$ This is essentially the same as the standard quadratic formula, just with different variable names. The reduced form is sometimes easier to work with because the leading coefficient is always 1, making calculations simpler. The Discriminant and Nature of Roots Why the discriminant matters: The discriminant $D = b^2 - 4ac$ is a single number that tells you everything about what type of solutions your quadratic has, without actually solving it. The three cases: If $D > 0$: The equation has two distinct real roots. This means the parabola crosses the $x$-axis at two different points. When you compute $\sqrt{D}$ in the quadratic formula, you get a positive number, and the $\pm$ gives you two different solutions. If $D = 0$: The equation has one real double root (also called a repeated root). This means the parabola just touches the $x$-axis at exactly one point. The quadratic formula gives $x = \frac{-b}{2a}$ (no $\pm$ because the square root term is zero), so you get one solution with multiplicity 2. If $D < 0$: The equation has two complex conjugate roots. This means the parabola doesn't touch the $x$-axis at all. When you try to take the square root of a negative number, you get imaginary numbers, resulting in a pair of complex solutions of the form $\alpha + \beta i$ and $\alpha - \beta i$. Key distinction: Roots are distinct (different from each other) precisely when $D \neq 0$. Roots are real precisely when $D \geq 0$. Example: Consider $x^2 - 4x + 3 = 0$. The discriminant is $D = 16 - 12 = 4 > 0$, so we expect two distinct real roots. Indeed, factoring gives $(x-1)(x-3) = 0$, so $x = 1$ or $x = 3$. Now consider $x^2 - 4x + 4 = 0$. The discriminant is $D = 16 - 16 = 0$, so we expect one double root. This factors as $(x-2)^2 = 0$, giving $x = 2$ (with multiplicity 2). The image above shows different parabolas and their roots. Notice how the shape and position of each parabola relates to its discriminant and the nature of its roots. <extrainfo> Geometric Constructions Beyond the algebraic methods, quadratic equations can be solved using geometric constructions. The Carlyle Circle (also called a Carlyle diagram) is a classical geometric method where a specially constructed circle intersects a line at points whose coordinates are the roots of the quadratic equation. While elegant and historically interesting, this method is rarely used in practice since algebraic methods are much more efficient. </extrainfo>