Welcome to our explanation of quadratic equations. A quadratic equation is a polynomial equation where the highest power of the variable is 2. The standard form of a quadratic equation is a x squared plus b x plus c equals zero, where a is the quadratic coefficient, b is the linear coefficient, and c is the constant term. When graphed, a quadratic equation forms a parabola, as shown in this example: y equals x squared plus 2x plus 3.
Now let's look at how to solve quadratic equations. There are two main methods. The first method is factoring. For example, to solve x squared plus 5x plus 6 equals zero, we factor it as (x plus 2) times (x plus 3) equals zero. This gives us the solutions x equals negative 2 or x equals negative 3. The second method is using the quadratic formula: x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. On the graph, we can see that the roots of the equation, or the x-intercepts of the parabola, are at x equals negative 2 and x equals negative 3.
The discriminant is a key part of the quadratic formula that tells us about the nature of the solutions. The discriminant is calculated as b squared minus 4ac. When the discriminant is greater than zero, the quadratic equation has two distinct real solutions, as shown by the blue parabola which crosses the x-axis at two points. When the discriminant equals zero, there is exactly one real solution, which is a repeated root. This is illustrated by the green parabola that touches the x-axis at exactly one point. When the discriminant is less than zero, there are no real solutions, only complex ones. The purple parabola doesn't intersect the x-axis at all, indicating no real solutions.
Another method for solving quadratic equations is completing the square. Let's solve 2x squared minus 12x plus 10 equals 0. First, we divide everything by 2 to get x squared minus 6x plus 5 equals 0. Next, we move the constant term to the right side: x squared minus 6x equals negative 5. To complete the square, we take half the coefficient of x, which is negative 3, square it to get 9, and add it to both sides. This gives us x squared minus 6x plus 9 equals negative 5 plus 9, which simplifies to (x minus 3) squared equals 4. Taking the square root of both sides, we get x minus 3 equals plus or minus 2, so x equals 1 or 5. On the graph, we can see the parabola crosses the x-axis at these two points, and its vertex is at the point (3, -4).
Quadratic equations have many real-world applications. One important application is in projectile motion, where the height of an object over time follows a quadratic function. The formula h of t equals negative one-half g t squared plus v-naught t plus h-naught describes the height of an object, where g is gravity, v-naught is initial velocity, and h-naught is initial height. In our example, a ball is thrown upward, reaches its maximum height at time t equals 2 seconds, and returns to the ground at t equals 4 seconds. Quadratic equations are also used in area problems, where we need to find dimensions with certain constraints. In economics, they appear in revenue, cost, and profit functions. Overall, quadratic equations are fundamental tools in mathematics, physics, engineering, and economics, helping us model and solve a wide range of real-world problems.