How to Solve Quadratic Equations Step by Step (All Methods Explained)

Quick Answer: How to Solve a Quadratic Equation

The fastest method depends on the equation:

• Factorable equations → solve by factoring

• Any equation → use the quadratic formula

• Missing b term (ax² + c = 0) → solve by square roots

• When asked to show your work fully → complete the square

The quadratic formula works for every quadratic equation, every time. It is the one method worth memorising.

Method 1: Solving Quadratic Equations by Factoring

Factoring works when the equation can be broken into two brackets that multiply to give the original equation.

When to use it:

• The equation has small, whole number coefficients

• You can spot the factor pairs quickly

Steps:

1. Write the equation in standard form: ax² + bx + c = 0

2. Find two numbers that multiply to a × c and add to b

3. Split the middle term using those two numbers

4. Factor by grouping

5. Set each bracket equal to zero and solve for x

Example: x² + 5x + 6 = 0

1. Find two numbers that multiply to 6 and add to 5 → 2 and 3

2. Rewrite: (x + 2)(x + 3) = 0

3. Set each factor to zero:

   • x + 2 = 0 → x = −2

   • x + 3 = 0 → x = −3

Answer: x = −2 or x = −3

Example with a leading coefficient: 2x² + 7x + 3 = 0

1. Multiply a × c = 2 × 3 = 6. Find two numbers that multiply to 6 and add to 7 → 1 and 6

2. Rewrite: 2x² + x + 6x + 3 = 0

3. Factor by grouping: x(2x + 1) + 3(2x + 1) = 0

4. Factor out: (x + 3)(2x + 1) = 0

5. Solve: x = −3 or x = −½

Method 2: How to Use the Quadratic Formula

The quadratic formula solves any quadratic equation regardless of whether it factors neatly. This is the most reliable method and the one you should default to when factoring is not obvious.

The Formula:

x = (−b ± √(b² − 4ac)) / 2a

Where a, b, and c come from the standard form ax² + bx + c = 0.

Steps:

1. Write the equation in standard form

2. Identify a, b, and c

3. Substitute into the formula

4. Calculate the discriminant: b² − 4ac

5. Solve for both values of x using + and −

Example: 2x² − 4x − 6 = 0

1. Identify: a = 2, b = −4, c = −6

2. Discriminant: (−4)² − 4(2)(−6) = 16 + 48 = 64

3. √64 = 8

4. x = (4 + 8) / 4 = 3

5. x = (4 − 8) / 4 = −1

Answer: x = 3 or x = −1

What the discriminant tells you:

When you are stuck on a problem and cannot spot the factors, plugging values into the quadratic formula is always the correct move. This is exactly what an AI math solver does when you scan a quadratic equation — it identifies the method, applies the formula, and shows every substitution step.

Method 3: Solving by Completing the Square

Completing the square is a technique that rewrites the equation into the form (x + p)² = q, which you can then solve by taking the square root.

When to use it:

• Required by your teacher or exam question

• The leading coefficient is 1

• Useful for deriving the quadratic formula itself

Steps:

1. Move the constant to the right side: x² + bx = −c

2. Take half of b, square it, and add to both sides

3. Write the left side as a perfect square

4. Take the square root of both sides

5. Solve for x

Example: x² + 6x + 5 = 0

1. Move constant: x² + 6x = −5

2. Half of 6 is 3. Square it: 9. Add to both sides: x² + 6x + 9 = 4

3. Write as square: (x + 3)² = 4

4. Take square root: x + 3 = ±2

5. Solve: x = −1 or x = −5

Method 4: Solving Quadratic Equations by Graphing

Graphing gives a visual answer. The solutions are the points where the parabola crosses the x-axis (the x-intercepts).

When to use it:

• You need to visualise the equation

• You are checking solutions from another method

• The question asks you to identify roots graphically

Steps:

1. Write in standard form: y = ax² + bx + c

2. Plot the parabola or use a graphing tool

3. Read the x-intercepts — these are your solutions

What the graph tells you:

• Two x-intercepts → two real solutions

• One x-intercept → one repeated solution

• No x-intercepts → no real solutions (discriminant is negative)

Graphing is useful for checking, but it is not precise for non-integer answers. The quadratic formula gives the exact values every time.

How to Solve a Quadratic Equation on a Calculator

For any quadratic, a scientific calculator handles the arithmetic once you have identified a, b, and c. The steps:

1. Calculate the discriminant: b² − 4ac

2. Take the square root of the result

3. Apply the formula: (−b + √discriminant) / 2a for the first root

4. Repeat with minus for the second root

If you want full working shown automatically, Calculator Air takes the equation as input, typed or photographed, and breaks down every step with the reasoning behind each move. Built for students who need to understand the method, not just read the answer.

Choosing the Right Method

Common Mistakes to Avoid

• Forgetting ± in the quadratic formula

  There are almost always two solutions. Missing the negative root loses marks.

  
  

• Sign errors with negative b

  The formula uses −b, so if b is already negative, −b becomes positive. Write it out carefully.

  
  

• Not setting the equation to zero first

  The formula only works when the equation is in standard form ax² + bx + c = 0. Rearrange before substituting.

  
  

• Stopping at the discriminant

  Students sometimes calculate b² − 4ac and stop. Keep going — that is only the middle step.

  
  

• Dividing only part of the numerator by 2a

  The entire expression (−b ± √discriminant) is divided by 2a, not just the square root part.

Frequently Asked Questions

1. What is the easiest way to solve a quadratic equation?

   The quadratic formula is the most reliable method because it works for any quadratic, whether it factors or not. For equations that factor quickly, factoring is faster.

   
   

2. What are the four ways to solve quadratic equations?

   Factoring, the quadratic formula, completing the square, and graphing. The quadratic formula works for all four cases, so it is the safest method when you are unsure which applies.

   
   

3. How do you solve a quadratic equation step by step?

   Write the equation in standard form (ax² + bx + c = 0), identify a, b, and c, then substitute into the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. Calculate the discriminant first, then solve for both values of x.

   
   

4. How do you solve quadratic equations by factoring?

   Find two numbers that multiply to a × c and add to b. Use those numbers to split the middle term, factor by grouping, set each bracket to zero, and solve for x.

   
   

5. What does the discriminant tell you?

   If b² − 4ac is positive, there are two real solutions. If it equals zero, there is one repeated solution. If it is negative, there are no real solutions.

   
   

6. Can I solve a quadratic equation with a photo?

   Yes. Using the photo math solver in Calculator Air, you can snap a picture of any quadratic equation, printed or handwritten, and get a full step-by-step solution with the method and reasoning shown at every stage.

Quadratic equations come up constantly across algebra, calculus, and science. The students who handle them quickly are usually the ones who have the quadratic formula memorised and know how to check their discriminant before committing to a method. For the rest, a reliable math homework helper that shows full working is the next best thing.