A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is two. The standard form is a x squared plus b x plus c equals zero, where a is not equal to zero. This creates a parabolic curve when graphed.
The standard form of a quadratic equation is a x squared plus b x plus c equals zero. It has three components: the quadratic term a x squared, the linear term b x, and the constant term c. For example, in the equation two x squared plus three x minus five equals zero, a equals two, b equals three, and c equals negative five.
There are three main methods to solve quadratic equations. First is factoring, where we express the equation as a product of two binomials. For example, x squared minus five x plus six equals zero factors to x minus two times x minus three equals zero, giving solutions x equals two and x equals three. The second method is the quadratic formula, and the third is completing the square.
The quadratic formula is x equals negative b plus or minus the square root of b squared minus four a c, all divided by two a. The discriminant, b squared minus four a c, determines the number of solutions. If the discriminant is positive, there are two real solutions. If it equals zero, there is one solution. If negative, there are no real solutions. For example, solving x squared minus four x plus three equals zero gives us x equals three and x equals one.
To summarize what we have learned about quadratic equations: They have the standard form a x squared plus b x plus c equals zero. There are three main methods to solve them: factoring, using the quadratic formula, and completing the square. The discriminant tells us how many real solutions exist. When graphed, quadratic equations form parabolas, and the solutions are the points where the parabola crosses the x-axis.