Introduction to Quadratic Equations

Definition and Graphs of Quadratic Functions What is a Quadratic Function? A quadratic function is a polynomial function of degree 2. The standard form is: $$y = ax^2 + bx + c$$ where $a$, $b$, and $c$ are real numbers and $a \neq 0$. The condition $a \neq 0$ is crucial—if $a = 0$, the equation becomes linear, not quadratic. When we set this function equal to zero, we get a quadratic equation: $$ax^2 + bx + c = 0$$ The key insight is that solving a quadratic equation means finding the values of $x$ where the function equals zero. The Graph: Understanding Parabolas When you graph a quadratic function, you always get a parabola—a symmetric, U-shaped or inverted U-shaped curve. This distinctive shape is a fundamental property of quadratic functions and appears nowhere else in basic algebra. The direction the parabola opens depends entirely on the coefficient $a$: When $a > 0$ (positive): The parabola opens upward ∪, creating a minimum point at the bottom When $a < 0$ (negative): The parabola opens downward ∩, creating a maximum point at the top The larger the absolute value $|a|$, the narrower (more stretched vertically) the parabola becomes. Roots, Zeros, and X-Intercepts The roots (or zeros) of a quadratic function are the $x$-values where the function equals zero. Geometrically, these are the points where the parabola crosses the $x$-axis. Why does this matter? When solving a quadratic equation like $2x^2 - 8x + 6 = 0$, you're finding exactly these points. The solutions you find are the $x$-intercepts of the parabola. This connection between algebra and geometry is powerful: if you can see the parabola crosses the $x$-axis at two points, you know there are two real solutions. Methods for Solving Quadratic Equations There are three main methods for solving quadratic equations. Understanding when to use each method is important. Factoring Method If a quadratic can be factored, this is often the fastest solution method. The idea is simple: rewrite the quadratic as a product of two linear factors. For example, if you can factor as: $$(px + q)(rx + s) = 0$$ then by the zero product property (if a product equals zero, at least one factor must equal zero), you get: $px + q = 0$ which gives $x = -\frac{q}{p}$ $rx + s = 0$ which gives $x = -\frac{s}{r}$ Example: Solve $x^2 + 5x + 6 = 0$ This factors as $(x + 2)(x + 3) = 0$, so $x = -2$ or $x = -3$. The limitation: not all quadratics factor nicely with integer coefficients, so you need other methods. Completing the Square Method This method transforms the quadratic into a perfect square, which can then be solved by taking square roots. The goal is to rewrite $ax^2 + bx + c = 0$ in the form $(x + d)^2 = e$. Example: Solve $x^2 + 6x + 5 = 0$ Move the constant: $x^2 + 6x = -5$ Complete the square: Take half of the $b$-coefficient ($6/2 = 3$) and square it ($3^2 = 9$) Add to both sides: $x^2 + 6x + 9 = -5 + 9$ Factor: $(x + 3)^2 = 4$ Take square roots: $x + 3 = \pm 2$ Solve: $x = -3 + 2 = -1$ or $x = -3 - 2 = -5$ This method always works, but it's more tedious than factoring when factoring is possible. The Quadratic Formula and Discriminant The Quadratic Formula Both factoring and completing the square methods, when applied generally to $ax^2 + bx + c = 0$, lead to a universal formula that works for any quadratic equation: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ This quadratic formula is your most powerful tool. When you encounter a quadratic that doesn't factor neatly, use this formula directly. Key points: The $\pm$ symbol means you get two solutions (usually) The solutions are $x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$ and $x = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$ You must memorize this formula Example: Solve $2x^2 + 3x - 2 = 0$ with $a = 2$, $b = 3$, $c = -2$ $$x = \frac{-3 \pm \sqrt{9 - 4(2)(-2)}}{2(2)} = \frac{-3 \pm \sqrt{9 + 16}}{4} = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4}$$ So $x = \frac{-3 + 5}{4} = \frac{1}{2}$ or $x = \frac{-3 - 5}{4} = -2$ The Discriminant The expression under the square root in the quadratic formula is called the discriminant: $$\Delta = b^2 - 4ac$$ The discriminant tells you crucial information about the nature and number of solutions before you even calculate them. Case 1: Discriminant Greater Than Zero ($\Delta > 0$) When the discriminant is positive, you're taking the square root of a positive number. This means: You get two distinct real solutions The parabola crosses the $x$-axis at two different points Example: For $x^2 - 5x + 6 = 0$, we have $\Delta = 25 - 24 = 1 > 0$, so there are two real solutions. Case 2: Discriminant Equal to Zero ($\Delta = 0$) When the discriminant is exactly zero: You get one real solution (technically a "double root" or "repeated root") The parabola touches the $x$-axis at exactly one point The $\pm$ in the formula becomes irrelevant since $\sqrt{0} = 0$ Example: For $x^2 - 4x + 4 = 0$, we have $\Delta = 16 - 16 = 0$, giving the single solution $x = 2$. Case 3: Discriminant Less Than Zero ($\Delta < 0$) When the discriminant is negative: You cannot take the square root of a negative number (in real numbers) The equation has no real solutions The parabola never crosses the $x$-axis The solutions are complex conjugates (pairs of complex numbers) Example: For $x^2 + 2x + 5 = 0$, we have $\Delta = 4 - 20 = -16 < 0$, so there are no real solutions. Why this matters: On an exam, if you compute a negative discriminant, you know immediately there are no real solutions—you don't need to continue with the formula. <extrainfo> Applications of Quadratic Equations Quadratic equations model many real-world situations: Projectile Motion When an object is thrown or launched, its height above the ground at time $t$ follows a quadratic equation: $$h(t) = -\frac{1}{2}gt^2 + v0 t + h0$$ where $g$ is gravitational acceleration, $v0$ is initial velocity, and $h0$ is initial height. Finding when $h(t) = 0$ tells you when the projectile hits the ground. Optimization Problems Many real-world problems ask for the maximum or minimum value (like maximum area with limited material, or minimum cost). Since parabolas have a highest or lowest point, quadratics naturally model these situations. Economic Modeling Revenue, cost, and profit functions in economics are often quadratic. For example, profit might be modeled as $P(x) = -2x^2 + 100x - 500$, where finding the roots or vertex determines break-even points and optimal production levels. </extrainfo>