Quadratic Formula Explainer with Simple Graph Animations

We'll solve x squared plus 5x plus 6 equals zero using the quadratic formula.First, let's recall the quadratic formula and plug in our values: a equals 1, b equals 5, and c equals 6.Let's start by calculating b squared, which is 5 squared equals 25.Next, we calculate 4ac, which is 4 times 1 times 6, equals 24.Now we can subtract: 25 minus 24 equals 1.The square root of 1 is simply 1.Now we can find both solutions. For the positive case, negative 5 plus 1, divided by 2, equals negative 2.And for the negative case, negative 5 minus 1, divided by 2, equals negative 3.These solutions can be verified by plugging them back into the original equation.Now let's visualize our quadratic equation y equals x squared plus five x plus six on a coordinate plane.As we draw the parabola, notice how it opens upward because the coefficient of x squared is positive.The x-intercepts are the points where the parabola crosses the x-axis. These are our solutions from the quadratic formula: negative three and negative two.At these x-intercepts, y equals zero. This is why finding the roots of a quadratic equation means finding where the parabola intersects the x-axis.Let's zoom in on these x-intercepts to get a closer look at where the parabola crosses the x-axis.Now, let's look at a different quadratic equation that doesn't cross the x-axis at all.When a parabola never touches or crosses the x-axis, it means the quadratic equation has imaginary solutions, because there are no real x-values that make y equal zero.