How to Solve Quadratic Equations Step by Step

A quadratic equation is any equation that can be written in the form ax squared plus bx plus c equals zero, where a is not zero. These equations describe everything from the path of a thrown ball to the shape of a satellite dish, which is why they sit at the heart of algebra. Solving one means finding the values of x that make the equation true, known as the roots or solutions. There are three main methods: factoring, completing the square, and the quadratic formula. This guide walks through each one with worked examples so you can choose the right tool for any problem.

Recognizing the Standard Form

Before solving, write your equation in standard form so that everything is on one side and zero is on the other. In ax squared plus bx plus c equals zero, the coefficient a multiplies x squared, b multiplies x, and c is the constant. For example, in 2x squared plus 3x minus 5 equals 0, we have a equals 2, b equals 3, and c equals negative 5. Identifying these three numbers correctly is essential because every method depends on them.

Method 1: Solving by Factoring

Factoring is the fastest method when it works. The idea is to rewrite the quadratic as a product of two binomials, then use the fact that if two things multiply to zero, at least one of them must be zero.

Example: Solve x squared plus 5x plus 6 equals 0.

1. Find two numbers that multiply to 6 (the constant) and add to 5 (the middle coefficient). Those numbers are 2 and 3.
2. Write the factored form: (x plus 2)(x plus 3) equals 0.
3. Set each factor equal to zero: x plus 2 equals 0 or x plus 3 equals 0.
4. Solve each: x equals negative 2 or x equals negative 3.

Check the answer by substituting. Plugging x equals negative 2 into the original gives 4 minus 10 plus 6, which equals 0. The solution works. Factoring is ideal when the numbers are clean whole numbers, but not every quadratic factors nicely, which is where the other methods come in.

Method 2: The Quadratic Formula

The quadratic formula solves every quadratic equation, even ones that cannot be factored. It states:

x equals negative b plus or minus the square root of (b squared minus 4ac), all divided by 2a.

Example: Solve 2x squared plus 3x minus 5 equals 0, where a equals 2, b equals 3, and c equals negative 5.

1. Calculate the discriminant, b squared minus 4ac: 3 squared is 9, and 4 times 2 times negative 5 is negative 40, so 9 minus negative 40 equals 49.
2. Take the square root of 49, which is 7.
3. Apply the formula: x equals (negative 3 plus or minus 7) divided by (2 times 2), which is (negative 3 plus or minus 7) divided by 4.
4. Compute both solutions. Using plus: (negative 3 plus 7) divided by 4 equals 4 divided by 4, which is 1. Using minus: (negative 3 minus 7) divided by 4 equals negative 10 divided by 4, which is negative 2.5.

So the solutions are x equals 1 and x equals negative 2.5. The formula never fails, making it the most reliable method to memorize.

Understanding the Discriminant

The expression under the square root, b squared minus 4ac, is called the discriminant, and it tells you what kind of solutions to expect before you finish solving.

• If the discriminant is positive, there are two different real solutions.
• If it equals zero, there is exactly one real solution (a repeated root).
• If it is negative, there are no real solutions, only complex ones, because you cannot take the real square root of a negative number.

In the example above the discriminant was 49, a positive number, which is why we found two distinct real roots.

Method 3: Completing the Square

Completing the square rewrites the equation so one side is a perfect square. It is especially useful for deriving the vertex of a parabola and for understanding where the quadratic formula comes from.

Example: Solve x squared plus 6x plus 5 equals 0.

1. Move the constant to the right: x squared plus 6x equals negative 5.
2. Take half of the middle coefficient (half of 6 is 3) and square it (3 squared is 9). Add 9 to both sides: x squared plus 6x plus 9 equals negative 5 plus 9.
3. The left side is now a perfect square: (x plus 3) squared equals 4.
4. Take the square root of both sides: x plus 3 equals plus or minus 2.
5. Solve: x equals negative 3 plus 2 equals negative 1, or x equals negative 3 minus 2 equals negative 5.

The solutions are x equals negative 1 and x equals negative 5, which you could verify by factoring as well.

Which Method Should You Use?

• Try factoring first when the coefficients are small whole numbers. It is the quickest path when it works.
• Use the quadratic formula when factoring is not obvious or the numbers are messy. It always gives an answer.
• Use completing the square when you need the vertex form or are asked to show the structure of the equation.

If you want to check your work or see each step laid out, our free math tools can solve quadratics and verify your roots. Strengthening your algebra foundation first also helps; see our guide on how to understand algebra easily for the prerequisite skills.

Common Mistakes to Avoid

• Forgetting the plus or minus. Taking a square root always yields two values, so most quadratics have two solutions.
• Sign errors with c. When c is negative, be careful with the negative 4ac term; two negatives make a positive.
• Not setting the equation to zero first. Every method assumes standard form with zero on one side.
• Dropping the coefficient a. Always divide by 2a in the formula, not just 2.

Reinforce these methods with regular practice problems so the steps become second nature.

Frequently Asked Questions

What is the easiest way to solve a quadratic equation?

If it factors cleanly, factoring is fastest. Otherwise the quadratic formula is the most dependable because it works for every quadratic regardless of the coefficients.

Can every quadratic equation be factored?

No. Many quadratics do not factor into whole numbers. For those, use the quadratic formula or completing the square to find the exact solutions.

What does it mean when there are no real solutions?

It means the discriminant is negative and the parabola never crosses the x axis. The solutions still exist but involve imaginary numbers, which appear in more advanced courses.

Why do quadratic equations usually have two answers?

Because a parabola can cross the x axis at two points. The plus or minus in the square root step produces those two intersection values.

Master these three methods and you can solve any quadratic that comes your way. Start with factoring, keep the quadratic formula ready for tougher problems, and use the discriminant to predict your results.