Step‑by‑Step Tutorial: How to Use the Quadratic Formula

What is a Quadratic Equation?

A quadratic equation is any equation that can be written in the standard form:

$a{x}^2+bx+c=0$

Here, a, b, and c are numerical coefficients, and a cannot be zero.

• a is the coefficient of the x2 term.

• b is the coefficient of the x term.

• c is the constant term.

For example, in the equation $2x^2+5x-3=0$, we have $a=2$, $b=5$, and $c=-3$.

The Quadratic Formula

The quadratic formula is the key to solving for x in any quadratic equation. It is:

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

This formula may look intimidating, but once you break it down, it's just a matter of plugging in your numbers and doing some basic arithmetic.

Step 1: Set Your Equation to Standard Form

Before you can use the formula, your quadratic equation must be in the standard form, $a{x}^2+bx+c=0$. If it's not, you need to rearrange it.

Example: Solve for x in $3x^2-6=-7x$.

1. Add 7x to both sides to move all terms to one side.

   3x2+7x−6=0

2. Now it's in standard form, and you can identify your coefficients:

   • $a=3$

   • $b=7$

   • $c=-6$

Step 2: Identify and Plug Your Coefficients into the Formula

Now, take the values you found for a, b, and c and substitute them into the quadratic formula. Be very careful with signs (positive and negative)!

Using our example, with $a=3$, $b=7$, and $c=-6$:

$x=\frac{-(7)\pm \sqrt{(7{)}^2-4(3)(-6)}}{2(3)}$

Step 3: Simplify the Discriminant

The expression under the square root, $b^2-4ac$, is called the discriminant. It's a good idea to simplify this part first, as its value tells you a lot about the solutions.

• If $b^2-4ac>0$: There are two real, distinct solutions.

• If $b^2-4ac=0$: There is exactly one real solution.

• If $b^2-4ac<0$: There are no real solutions (the solutions are complex numbers).

Let's simplify the discriminant for our example:

b2−4ac=(7)2−4(3)(−6)

=49−4(−18)

=49+72

=121

Since 121 is greater than zero, we know we will have two real solutions.

Step 4: Solve for $x$

Now that you have simplified the discriminant, put it back into the formula and solve for x. The ± (plus-minus) sign means you have to calculate two separate solutions: one using the plus sign and one using the minus sign.

x=6−7±121​​

x=6−7±11​

Solution 1 (using the plus sign):

$x=\frac{-7+11}{6}=\frac{4}{6}=\frac{2}{3}$

Solution 2 (using the minus sign):

$x=\frac{-7-11}{6}=\frac{-18}{6}=-3$

So, the two solutions for the equation $3x^2+7x-6=0$ are $x=\frac{2}{3}$ and $x=-3$.

Tips for Success

• Double-check your signs: This is the most common mistake. A single negative sign in the wrong place can throw off your entire calculation.

• Simplify the discriminant first: This breaks the problem into a smaller, more manageable step.

• Remember the $\pm$: Don't forget to calculate both the "plus" and "minus" solutions.

Useful Tools

If you want to check your work or solve a problem quickly, these online calculators can be very helpful:

• Quadratic Formula Calculator: This tool shows the step-by-step solution process for any quadratic equation.

  • https://www.calcsphere.com/2025/08/quadratic-formula-calculator-solve.html

• Scientific Quadratic Equation Solver: A more streamlined solver that gives you the direct answer.

  • https://www.calcsphere.com/2025/08/scientific-quadratic-equation-solver.html

By following these steps, you can confidently solve any quadratic equation using the quadratic formula. Practice is key, so try a few more examples on your own!