A quadratic equation is a polynomial equation of the second degree. It is written in the standard form a x squared plus b x plus c equals zero, where a, b, and c are constants called coefficients, and x is the variable. Importantly, the coefficient a must be non-zero, otherwise the equation would not be quadratic. The graph of a quadratic function forms a parabola, as shown here with the simple example f of x equals x squared. A typical quadratic equation might look like x squared minus 4x plus 3 equals zero.
There are three main methods to solve quadratic equations. First, factoring, which works well when the equation can be easily factored. Second, completing the square, which is useful for converting to vertex form. Third, the quadratic formula, which works for any quadratic equation. The quadratic formula states that x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. Let's solve an example: x squared minus 5x plus 6 equals zero. First, we identify that a equals 1, b equals negative 5, and c equals 6. Substituting these values into the formula, we get x equals 5 plus or minus the square root of 25 minus 24, all divided by 2. This simplifies to 5 plus or minus 1, divided by 2. So x equals 3 or x equals 2. These are our two solutions.
The discriminant, denoted by delta, is a key part of the quadratic formula. It equals b squared minus 4ac. The discriminant tells us what type of solutions a quadratic equation has. If the discriminant is positive, the equation has two distinct real solutions. We can see this with the parabola y equals x squared minus 4, which crosses the x-axis at negative 2 and positive 2. If the discriminant equals zero, there is exactly one repeated real solution. This is shown by the parabola y equals x squared, which touches the x-axis at exactly one point, the origin. If the discriminant is negative, there are no real solutions, only complex ones. The parabola y equals x squared plus 4 never crosses the x-axis, as it's always above it.
Quadratic equations have many real-world applications. In physics, they model projectile motion, such as the path of a thrown ball. In economics, they help optimize profit by finding the price that maximizes revenue. Engineers use them to calculate areas and volumes, while financial analysts apply them to compound interest problems. Let's look at a specific example from physics: projectile motion. When an object is thrown upward, its height h at time t can be modeled by the quadratic equation h of t equals negative 4.9 t squared plus v-zero t plus h-zero, where v-zero is the initial velocity and h-zero is the initial height. In our example, with an initial velocity of 30 meters per second and starting from ground level, the object reaches a maximum height of about 45.9 meters after 3.06 seconds, and returns to the ground after 6.12 seconds. This is a perfect example of how quadratic equations help us understand and predict real-world phenomena.
To summarize what we've learned about quadratic equations: First, a quadratic equation has the standard form a x squared plus b x plus c equals zero, where a is not equal to zero. Second, we can solve quadratic equations using three main methods: factoring, completing the square, or the quadratic formula. Third, the discriminant, which is b squared minus 4ac, tells us whether the equation has two distinct real solutions, one repeated real solution, or two complex solutions. Fourth, quadratic equations have numerous real-world applications in fields like physics, economics, engineering, and finance. Finally, understanding quadratic equations provides a foundation for higher mathematics and is essential for solving many practical problems. Quadratic equations are one of the most fundamental and useful mathematical tools across many disciplines.