A quadratic equation is a polynomial equation of the second degree in one variable. Its general form is a x squared plus b x plus c equals zero, where a, b, and c are constants, with a not equal to zero, and x is the variable. The graph of a quadratic function forms a parabola. When we set the quadratic expression equal to zero, the solutions, or roots, represent the x-coordinates where the parabola crosses the x-axis.
There are three main methods to solve quadratic equations. First, the factoring method, where we rewrite the equation as a product of linear factors. Second, completing the square, which transforms the equation into the form (x plus h) squared equals k. Third, the quadratic formula, which gives us the solutions directly: x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. The expression b squared minus 4ac is called the discriminant. When the discriminant is positive, the equation has two distinct real roots, as shown by the blue parabola. When it equals zero, there is exactly one real root, as shown by the green parabola. And when it's negative, there are no real roots, as shown by the red parabola that doesn't cross the x-axis.
Let's look at the factoring method for solving quadratic equations. First, ensure the equation is in standard form with zero on one side. Then, factor the quadratic expression into a product of two linear factors. Next, apply the zero product property, which states that if a product equals zero, at least one of the factors must be zero. This gives us two simple linear equations to solve. Let's see an example: solve x squared plus 5x plus 6 equals zero. We factor the left side to get (x plus 2) times (x plus 3) equals zero. By the zero product property, either x plus 2 equals zero or x plus 3 equals zero. Solving these equations gives us x equals negative 2 or x equals negative 3. We can verify these are the roots by looking at where the parabola crosses the x-axis.
The quadratic formula is a powerful method that works for any quadratic equation. For an equation in the form ax squared plus bx plus c equals zero, the solutions are given by x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. The expression under the square root, b squared minus 4ac, is called the discriminant. If the discriminant is positive, we get two distinct real roots. If it's zero, we get one repeated real root. And if it's negative, we get two complex conjugate roots. Let's solve the equation 2x squared minus 4x minus 3 equals zero. We identify that a equals 2, b equals negative 4, and c equals negative 3. The discriminant equals b squared minus 4ac, which is 16 plus 24, giving us 40. Since the discriminant is positive, we have two real roots. Substituting into the formula, we get x equals 4 plus or minus the square root of 40, all divided by 4. Simplifying, we get x equals 1 plus or minus the square root of 10 divided by 2, which gives us approximately 2.58 and negative 0.58.
To summarize what we've learned about quadratic equations: A quadratic equation has the standard form a x squared plus b x plus c equals zero, where a is not equal to zero. We can solve these equations using three main methods: factoring, completing the square, and the quadratic formula. The discriminant, which is b squared minus 4ac, tells us about the nature of the roots. When graphed, a quadratic equation forms a parabola, and the roots are the x-coordinates where the parabola crosses the x-axis. Quadratic equations have numerous real-world applications in physics, engineering, economics, and many other fields. Understanding how to solve these equations is a fundamental skill in mathematics that will serve as a foundation for more advanced topics.