Algebra · Step-by-step guide

The Quadratic Formula

Use the quadratic formula accurately, interpret the discriminant, and simplify real or complex solutions.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

Write the equation as ax2+bx+c=0\displaystyle ax^2+bx+c=0, identify a, b, and c with their signs, and substitute into x=(−b±b2−4ac)/(2a)\displaystyle x=(-b\pm\sqrt{b^2-4ac})/(2a).

x=−b±b2−4ac2a\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Standard form first

All terms must be on one side before you identify coefficients. A missing term has coefficient 0.

2

Use the discriminant

b2−4ac\displaystyle b^2-4ac predicts two, one, or no real solutions before you finish.

3

Simplify carefully

Keep the entire numerator over 2a and reduce radicals and fractions at the end.

02

See the structure

A picture makes the relationships easier to remember.

−22vertex (0, −1)
The coordinate graph marks the vertex and the two x-intercepts of a representative quadratic.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Two rational roots

Solve x2−5x+6=0\displaystyle x^2-5x+6=0.

  1. a=1,b=−5,c=6\displaystyle a=1,b=-5,c=6.
  2. The discriminant is 25−24=1\displaystyle 25-24=1.
  3. x=(5±1)/2\displaystyle x=(5\pm1)/2.
Answerx=2,3\displaystyle x=2,3
Example 2Irrational roots

Solve 2x2+3x−1=0\displaystyle 2x^2+3x-1=0.

  1. a=2,b=3,c=−1\displaystyle a=2,b=3,c=-1.
  2. The discriminant is 9+8=17\displaystyle 9+8=17.
  3. Substitute and keep the radical exact.
Answerx=(−3±17)/4\displaystyle x=(-3\pm\sqrt{17})/4
Example 3One solution

Solve x2+4x+4=0\displaystyle x^2+4x+4=0.

  1. The discriminant is 16−16=0\displaystyle 16-16=0.
  2. Both formula branches give the same value.
  3. This is a repeated root.
Answerx=−2\displaystyle x=-2
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Dropping the sign of b or c.

  • Putting only the radical over 2a.

  • Using ± nowhere—or using it twice.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Quadratic Formula Practice

Practice coefficient identification, discriminants, and exact solutions.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Solve x2−5x+6=0\displaystyle x^2-5x+6=0.

Show hint

Use a=1,b=−5,c=6\displaystyle a=1,b=-5,c=6.

Show worked solution
  1. x=(5±25−24)/2\displaystyle x=(5\pm\sqrt{25-24})/2.
  2. x=(5±1)/2\displaystyle x=(5\pm1)/2.

Answer: x=2\displaystyle x=2 or x=3\displaystyle x=3.

2

Solve 2x2+x−3=0\displaystyle 2x^2+x-3=0.

Show hint

Substitute carefully into the formula.

Show worked solution
  1. x=(−1±1+24)/4\displaystyle x=(-1\pm\sqrt{1+24})/4.
  2. x=(−1±5)/4\displaystyle x=(-1\pm5)/4.

Answer: x=1\displaystyle x=1 or x=−3/2\displaystyle x=-3/2.

3

Solve x2+4x+8=0\displaystyle x^2+4x+8=0.

Show hint

The discriminant is negative.

Show worked solution
  1. x=(−4±16−32)/2\displaystyle x=(-4\pm\sqrt{16-32})/2.
  2. −16=4i\displaystyle \sqrt{-16}=4i.
  3. Simplify.

Answer: x=−2±2i\displaystyle x=-2\pm2i.

4

How many real solutions does 3x2−2x+5=0\displaystyle 3x^2-2x+5=0 have?

Show hint

Compute the discriminant.

Show worked solution
  1. b2−4ac=(−2)2−4(3)(5)\displaystyle b^2-4ac=(-2)^2-4(3)(5).
  2. 4−60=−56<0\displaystyle 4-60=-56<0.
  3. A negative discriminant gives no real roots.

Answer: No real solutions.

5

Solve 4x2−12x+9=0\displaystyle 4x^2-12x+9=0.

Show hint

Look for a zero discriminant.

Show worked solution
  1. x=(12±144−144)/8\displaystyle x=(12\pm\sqrt{144-144})/8.
  2. The square-root term is zero.
  3. x=12/8\displaystyle x=12/8.

Answer: x=3/2\displaystyle x=3/2 (double root).

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

Write the equation as ax2+bx+c=0\displaystyle ax^2+bx+c=0, identify a, b, and c with their signs, and substitute into x=(−b±b2−4ac)/(2a)\displaystyle x=(-b\pm\sqrt{b^2-4ac})/(2a).

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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