The quadratic formula is a powerful tool for solving quadratic equations. It provides a systematic approach that can be used to find the roots of any quadratic equation in the form ax^2 + bx + c = 0. In this article, we’ll break down the process into eight steps to help you understand and derive the quadratic formula easily.

Step 1: Write down the general form of a quadratic equation.

A quadratic equation is an equation of degree 2, and its general form is:

ax^2 + bx + c = 0

Here, a, b, and c are constants, and x is the variable.

Step 2: Use the technique called “completing the square.”

To derive the quadratic formula, we’ll employ a technique known as “completing the square,” which involves rewriting our equation in perfect square trinomial form. This will make it easier to solve for x.

Step 3: Divide both sides by ‘a.’

To begin completing the square, divide both sides of your equation by ‘a’ so that your leading coefficient (the coefficient of x^2) is 1:

x^2 + (b/a)x + (c/a) = 0

Step 4: Isolate x terms on one side.

Now, move the constant term (c/a) to the other side of your equation:

x^2 + (b/a)x = -(c/a)

Step 5: Complete the square.

Add a value ‘k’ on both sides of your equation that will complete the square on the left-hand side:

x^2 + (b/a)x + k = -(c/a) + k

The value of k is equal to half of (b/a)^2:

k = ((b/2a))^2

Step 6: Re-write your equation in perfect square trinomial form.

Your equation should now be in perfect square trinomial form:

(x + b/2a)^2 = b^2/(4a^2) – (c/a)

Step 7: Solve for ‘x.’

Take the square root of both sides of your equation:

x + b/(2a) = ± √(b^2 – 4ac) / (2a)

Now, isolate x by moving the term ‘b/(2a)’ to the other side:

x = -b/(2a) ± √(b^2 – 4ac) / (2a)

Step 8: Simplify.

Finally, simplify your expression by combining the parts with the common denominator:

x = (-b ± √(b^2 – 4ac)) / (2a)

Congratulations! You’ve derived the quadratic formula. Now, you can use this formula to solve any quadratic equation in the form ax^2 + bx + c = 0. Simply plug in the coefficients ‘a’, ‘b’, and ‘c’ into the formula and solve for ‘x.’