Quadratic Formula Explainer with Simple Graph Animations

The quadratic formula comes from the standard form of a quadratic equation.In the standard form, we have three important components.These components are used in the quadratic formula to find the values of x that make the equation equal to zero.Let's see how each component appears in the formula.Let's look at a specific example: x squared plus five x plus six equals zero.In this equation, we can identify our three components.A equals one, the coefficient of x squared, though it's not written explicitly.B equals five, the coefficient of x.And C equals six, our constant term.These values will be substituted into the quadratic formula to solve for x.Let's substitute our values into the quadratic formula.First, we plug in a equals 1, b equals 5, and c equals 6.Next, we simplify inside the square root. Five squared is twenty-five, and four times one times six is twenty-four.Twenty-five minus twenty-four equals one under the square root.The square root of one is simply one.Let's solve the positive case first. Negative five plus one equals negative four.Simplifying negative four divided by two...Now for the negative case. Negative five minus one equals negative six.Simplifying negative six divided by two gives us negative three.Here are some helpful tips for keeping track of negative signs.Remember that negative five plus one equals negative four, and negative five minus one equals negative six.Therefore, our solutions are x equals negative two and x equals negative three.Now that we've found our solutions algebraically, let's verify them graphically.Here's our parabola y equals x squared plus 5x plus 6.The x-intercepts are the points where our parabola crosses the x-axis. These are our solutions: negative 2 and negative 3.The discriminant, b squared minus 4ac, tells us about the number of solutions a quadratic equation has.The coefficient 'a' determines whether the parabola opens upward or downward.When a is positive, like in our example where a equals 1, the parabola opens upward.When a is negative, the parabola opens downward, but still crosses at the same x-intercepts.These x-intercepts remain the same regardless of the value of a, as long as we're working with the same base equation.