Solve for X: Quadratic Formula Made Easy

Welcome to understanding the quadratic formula! This powerful tool helps us solve quadratic equations.The quadratic formula is used to find the values of x that satisfy any quadratic equation.Before we use this formula, we need our equation in standard form: a x squared plus b x plus c equals zero.Let's understand what a, b, and c represent in a quadratic equation.Let's look at some examples to practice identifying these coefficients.In our first example, x squared plus five x plus six equals zero, we can identify that a equals one, b equals five, and c equals six.In our second example, two x squared minus three x minus one equals zero, a is two, b is negative three, and c is negative one.And in our final example, three x squared plus six x minus two equals zero, we have a equals three, b equals six, and c equals negative two.Here are some important tips for identifying coefficients: If no coefficient is written, it equals one. Always include the sign, and make sure your equation is in standard form.Now that we can identify a, b, and c, we're ready to use these values in the quadratic formula.Let's substitute our values into the quadratic formula.We'll use the quadratic formula and carefully substitute a equals 1, b equals 5, and c equals 6.First, let's substitute these values directly into the formula.Now, let's follow the proper order of operations to simplify this expression.We start by calculating b squared, which is 5 squared, giving us 25.Next, we calculate 4 times a times c, which is 4 times 1 times 6, giving us 24.Under the square root, we subtract 24 from 25, giving us 1.The square root of 1 is simply 1.The plus-minus symbol means we'll actually get two different answers when we complete our calculation.Now let's complete our calculation to find both solutions.Under the square root, we have twenty-five minus twenty-four, which equals one.Taking the square root of one gives us plus or minus one.For the positive case, we get negative five plus one, divided by two, which equals negative two.For the negative case, we get negative five minus one, divided by two, which equals negative three.Let's verify these solutions by plugging them back into our original equation.For x equals negative two, let's substitute into x squared plus five x plus six.Similarly, for x equals negative three, we can verify it's also a solution.The number of solutions a quadratic equation has depends on its discriminant, which is b squared minus four a c.When the discriminant is positive, like in our example, we get two different real solutions.When the discriminant equals zero, we get exactly one solution, called a repeated root.When the discriminant is negative, we get no real solutions, only complex solutions.