Welcome to our lesson on solving quadratic equations. A quadratic equation has the form a x squared plus b x plus c equals zero, where a is not equal to zero. There are three main methods to solve quadratic equations: factoring, completing the square, and using the quadratic formula. When we solve a quadratic equation, we're finding the x-values where the parabola crosses the x-axis, which are called the roots or solutions of the equation. For example, the function f of x equals x squared minus 2x minus 3 has roots at x equals negative 1 and x equals 3.
Let's look at our first method: solving by factoring. To solve a quadratic equation by factoring, first move all terms to one side to get the standard form. Then factor the expression into a product of linear factors. Next, set each factor equal to zero and solve the resulting linear equations. For example, to solve x squared minus 2x minus 3 equals zero, we factor the left side to get (x minus 3) times (x plus 1) equals zero. Using the zero product property, either x minus 3 equals zero, which gives us x equals 3, or x plus 1 equals zero, which gives us x equals negative 1. These are the x-intercepts of the parabola, where the graph crosses the x-axis.
Our second method is completing the square. This method is useful when the quadratic expression cannot be easily factored. First, move the constant term to the right side of the equation. If needed, divide all terms by the coefficient of x squared to make it 1. Then add the square of half the coefficient of x to both sides. This allows us to rewrite the left side as a perfect square trinomial. Next, take the square root of both sides, which gives us two possible values. Finally, solve for x. For example, to solve x squared minus 6x plus 5 equals zero, we rearrange to get x squared minus 6x equals negative 5. Adding 9 to both sides gives us x squared minus 6x plus 9 equals 4, which can be written as (x minus 3) squared equals 4. Taking the square root, we get x minus 3 equals plus or minus 2, so x equals 1 or 5. Geometrically, completing the square helps us find the vertex of the parabola, which is at (3, -4) in this example.
Our third and most powerful method is the quadratic formula. For any quadratic 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, and it tells us about the nature of the solutions. If the discriminant is positive, we get two distinct real solutions. If it's zero, we get exactly one real solution. And if it's negative, we get two complex solutions. Let's solve 2x squared plus 4x minus 3 equals zero using the formula. We identify a equals 2, b equals 4, and c equals negative 3. Substituting into the formula, we get x equals negative 4 plus or minus the square root of 16 plus 24, all divided by 4. This simplifies to negative 1 plus or minus the square root of 10 divided by 2, which gives us approximately 0.58 or negative 2.58. The discriminant is 40, which is positive, confirming we have two real solutions.
Let's summarize what we've learned about solving quadratic equations. A quadratic equation has the form ax squared plus bx plus c equals zero, where a is not equal to zero. We explored three methods to solve these equations: factoring, completing the square, and using the quadratic formula. Factoring works best when the quadratic expression can be easily factored into a product of linear factors. Completing the square is useful for finding the vertex form of the parabola and works even when factoring is difficult. The quadratic formula is the most general method and always works for any quadratic equation. The discriminant in the formula tells us whether we'll have two real solutions, one real solution, or two complex solutions. By mastering these methods, you can solve any quadratic equation you encounter.