Welcome to quadratic equations! A quadratic equation is a polynomial equation of the second degree. The standard form is a x squared plus b x plus c equals zero, where a, b, and c are coefficients, and importantly, a cannot be zero. The graph of a quadratic equation is a parabola, which has a vertex and may intersect the x-axis at points called roots or solutions.
Let's break down the components of a quadratic equation. In the standard form a x squared plus b x plus c equals zero, we have three distinct parts. The quadratic term a x squared contains the coefficient a. The linear term b x contains the coefficient b. And the constant term c stands alone. For example, in 2 x squared plus 5 x minus 3 equals zero, a equals 2, b equals 5, and c equals negative 3. Remember, a must never be zero, or the equation becomes linear instead of quadratic.
The quadratic formula is the most powerful method for solving quadratic equations. It states that 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. The plus-minus symbol means we get two solutions. Let's see an example: for x squared minus 5x plus 6 equals zero, we substitute into the formula to get x equals 3 or x equals 2.
The discriminant, represented by delta, equals b squared minus 4ac. It tells us about the nature of the roots before we solve the equation. When the discriminant is positive, the parabola crosses the x-axis at two points, giving us two real roots. When it equals zero, the parabola touches the x-axis at exactly one point, giving us one repeated root. When the discriminant is negative, the parabola doesn't touch the x-axis at all, meaning there are no real roots.
Let's summarize the three main methods for solving quadratic equations. First, factoring works when the quadratic can be written as a product of two binomials. For example, x squared minus 5x plus 6 factors to x minus 2 times x minus 3, giving us x equals 2 or x equals 3. Second, the quadratic formula always works for any quadratic equation. Third, completing the square is useful for converting to vertex form. Choose your method based on the equation's structure. With practice, you'll master all these techniques and be able to solve any quadratic equation confidently!