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Quadratic Equation Calculator with Steps

Q: How does the discriminant tell you the number of solutions?

The discriminant \Delta = b^2 - 4ac sits under the square root. If \Delta > 0, you get two real roots. If \Delta = 0, \sqrt{0} = 0 collapses the \pm into one repeated root. If \Delta < 0, the square root of a negative gives an imaginary number, so you get two complex conjugate roots.

Q: How do you derive the quadratic formula with steps?

Start with ax^2 + bx + c = 0 and complete the square: divide by a to get x^2 + \frac{b}{a}x = -\frac{c}{a}; add \left(\frac{b}{2a}\right)^2 to both sides; factor the left side as \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}; take the square root of both sides; then subtract \frac{b}{2a} to arrive at x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Q: Why is −b positive if b is negative?

The minus sign in front of b means "the opposite of b". If b = -5, then -b = -(-5) = +5.

Type or snap a quadratic — get both roots with every step shown.

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Quadratic Equation Calculator with Steps: Free Quadratic Formula Solver

Math AI Solver breaks second-degree polynomial equations into clear, line-by-line proofs. Unlike scientific calculators that show only final numbers, it reveals every intermediate step — whether you solve by factoring, completing the square, or find imaginary roots.

Line-by-line proofs

Follow clear, step-by-step deductions from standard form to final roots.

Complex root support

Full handling of imaginary numbers (

i = \sqrt{-1}

) when the discriminant is negative.

Multiple solution methods

Ask for the quadratic formula, factoring, or completing the square — the solver explains the method it picks.

100% free & no sign-up

Unlimited access with zero account creation or subscription fees.

The method

How to Solve Quadratic Equations Step-by-Step

A quadratic equation calculator with steps solves second-degree polynomial equations (

ax^2 + bx + c = 0

) and displays the complete algebraic derivation from standard form to exact roots.

Standard form

Set the equation equal to 0
Order terms as $ax^2 + bx + c$
Identify $a$ , $b$ , and $c$

Discriminant test

Compute $\Delta = b^2 - 4ac$
$\Delta > 0$ : two real roots
$\Delta = 0$ : one real root
$\Delta < 0$ : two complex roots

Method & solution

Factoring
Completing the square
Quadratic formula
Vertex & zeros

Step 1

Convert to standard form

Move every term to the left so the right side is zero:

ax^2 + bx + c = 0

Order terms by descending degree:

ax^2

, then

bx

, then

c

Step 2

Identify a, b, and c

a

is the coefficient of

x^2

(

a \neq 0

b

the coefficient of

x

, and

c

the constant — watch the negative signs. In

2x^2 - 5x - 3 = 0

a = 2

b = -5

c = -3

Step 3

Choose the best method

Factoring: when integer factors of $c$ sum to $b$ .
Square root property: when $b = 0$ .
Completing the square: when $a = 1$ and $b$ is even.
Quadratic formula: works for every quadratic, including radical and complex roots.

Step 4

Review the line-by-line proof

Math AI Solver gives the exact solutions with a step-by-step explanation, so you can check your own homework line by line.

Formula sheet

The Quadratic Formula & The Discriminant

One universal formula solves any second-degree polynomial:

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

x

The roots, zeros, or x-intercepts of the equation.

-b

and

\pm

The opposite of the linear coefficient;

\pm

gives two roots

x_1

and

x_2

b^2 - 4ac

The discriminant

\Delta

— it decides how many and what kind of solutions.

2a

Divides the entire numerator, not just the square root.

Understanding the discriminant

Discriminant	Root type	Graph behavior	Example
$\Delta > 0$	Two distinct real roots	Parabola crosses the x-axis at two points	$x^2 - 4 = 0\;(\Delta = 16)$
$\Delta = 0$	One repeated real root	Vertex touches the x-axis at one point	$x^2 - 4x + 4 = 0\;(\Delta = 0)$
$\Delta < 0$	Two complex roots $(a \pm bi)$	Parabola stays entirely above or below the x-axis	$x^2 + 9 = 0\;(\Delta = -36)$

For the full derivation sheet, see the quadratic formula reference guide.

Four methods

Four Ways to Solve a Quadratic Equation

No single method suits every problem. Math AI Solver supports all four main algebraic techniques.

1. The quadratic formula

Works for every quadratic, whatever the coefficients.

Calculate the discriminant $\Delta = b^2 - 4ac$ .
Substitute into $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ .
Simplify the radical and evaluate both the $+$ and $-$ branches.

2. Factoring

Use when

ax^2 + bx + c

splits into integer binomials

(px + q)(rx + s) = 0

When $a > 1$ , use the AC method: find factors of $a \cdot c$ that sum to $b$ , split the middle term, and factor by grouping.
Apply the zero product property: set each factor equal to 0.
Solve each linear equation for $x$ .

3. Completing the square

Rewrites the equation in vertex form

(x + h)^2 = k

— best when

a = 1

and

b

is even.

Divide by $a$ : $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$ .
Move the constant: $x^2 + \frac{b}{a}x = -\frac{c}{a}$ .
Add $\left(\frac{b}{2a}\right)^2$ to both sides and factor the left side.
Take the square root of both sides and isolate $x$ .

4. Taking square roots

When there is no linear term (

b = 0

), skip the full formula.

Isolate the squared term: $ax^2 = -c \Rightarrow x^2 = -\frac{c}{a}$ .
Take the square root: $x = \pm\sqrt{-\frac{c}{a}}$ .

Worked examples

Worked Step-by-Step Examples

Example 1 · Quadratic formula

Solve

2x^2 - 5x - 3 = 0

Coefficients:

a = 2

b = -5

c = -3

.Discriminant:

\Delta = (-5)^2 - 4(2)(-3) = 25 + 24 = 49

A positive perfect square, so two rational roots.Substitute:

x = \frac{5 \pm 7}{4}

Final answer:

x = 3

x = -\frac{1}{2}

Example 2 · Complex roots

Solve

x^2 - 4x + 13 = 0

Coefficients:

a = 1

b = -4

c = 13

.Discriminant:

\Delta = 16 - 52 = -36

Negative, so the parabola never crosses the x-axis.Simplify:

\sqrt{-36} = 6i

, so

x = \frac{4 \pm 6i}{2} = 2 \pm 3i

Final answer:

x = 2 + 3i

x = 2 - 3i

Example 3 · Factoring

Solve

x^2 + 7x + 12 = 0

Find factors of 12 that sum to 7:

3 \cdot 4 = 12

and

3 + 4 = 7

.Factor:

(x + 3)(x + 4) = 0

Zero product property:

x + 3 = 0

x + 4 = 0

.Final answer:

x = -3

x = -4

Graphing

Finding the Vertex and Graphing the Parabola

Every quadratic function

y = ax^2 + bx + c

traces a U-shaped curve called a parabola. These are its key landmarks:

Vertex (h, k)

The turning point — a minimum when

a > 0

, a maximum when

a < 0

h = -\frac{b}{2a},\; k = f(h)

Axis of symmetry

The vertical line splitting the parabola into mirror halves:

x = -\frac{b}{2a}

x-intercepts

Where

y = 0

— the same roots the quadratic formula gives.

y-intercept

Where the curve crosses the vertical axis:

(0, c)

Graph vs. algebra: tools like Desmos plot curves interactively; Math AI Solver connects those curves to the line-by-line algebraic solution you write on paper.

Common Mistakes When Solving Quadratics

1
Sign errors with −b
If $b = -5$ , then $-b = -(-5) = +5$ . Forgetting to flip the sign is the most common quadratic mistake.
2
Dropping the ± symbol
Leaving out $\pm$ gives you only one root instead of both solutions.
3
Not dividing the whole numerator by 2a
$2a$ divides both $-b$ and the radical: $\frac{-b \pm \sqrt{\Delta}}{2a}$ . Never write $-b \pm \frac{\sqrt{\Delta}}{2a}$ .
4
Miscalculating −4ac with negatives
When $c$ is negative, $-4ac$ becomes positive: $-4(2)(-3) = +24$ , not $-24$ .
5
Squaring negatives incorrectly
With $b = -5$ , $(-5)^2 = +25$ . Typing $-5^2$ on a calculator gives $-25$ and breaks the discriminant.

FAQ

Frequently Asked Questions

Math AI Solver is a free quadratic formula calculator with steps. It computes real and complex solutions, works by factoring, completing the square, or the quadratic formula, and never hides intermediate steps behind a paywall.

The discriminant

\Delta = b^2 - 4ac

sits under the square root. If

\Delta > 0

, you get two real roots. If

\Delta = 0

\sqrt{0} = 0

collapses the

\pm

into one repeated root. If

\Delta < 0

, the square root of a negative gives an imaginary number, so you get two complex conjugate roots.

Yes. Math AI Solver evaluates negative radicands by factoring out

\sqrt{-1} = i

. Instead of reporting an error, it shows the solutions in standard complex form

a \pm bi

Start with

ax^2 + bx + c = 0

and complete the square: divide by

a

to get

x^2 + \frac{b}{a}x = -\frac{c}{a}

; add

\left(\frac{b}{2a}\right)^2

to both sides; factor the left side as

\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}

; take the square root of both sides; then subtract

\frac{b}{2a}

to arrive at

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

Use factoring when the expression easily breaks into integer factors whose sum equals

b

. If the equation has large numbers, fractions, or irrational or complex roots, the quadratic formula is the most reliable approach.

The minus sign in front of

b

means "the opposite of

b

". If

b = -5

, then

-b = -(-5) = +5

Quadratic Equation Calculator with Steps

Quadratic Equation Calculator with Steps: Free Quadratic Formula Solver

Line-by-line proofs

Complex root support

Multiple solution methods

100% free & no sign-up

How to Solve Quadratic Equations Step-by-Step

Standard form

Discriminant test

Method & solution

Convert to standard form

Identify a, b, and c

Choose the best method

Review the line-by-line proof

The Quadratic Formula & The Discriminant

Understanding the discriminant

Four Ways to Solve a Quadratic Equation

1. The quadratic formula

2. Factoring

3. Completing the square

4. Taking square roots

Worked Step-by-Step Examples

Finding the Vertex and Graphing the Parabola

Vertex (h, k)

Axis of symmetry

x-intercepts

y-intercept

Common Mistakes When Solving Quadratics

Sign errors with −b

Dropping the ± symbol

Not dividing the whole numerator by 2a

Miscalculating −4ac with negatives

Squaring negatives incorrectly

Frequently Asked Questions

What is the best free quadratic equation calculator with steps?

How does the discriminant tell you the number of solutions?

Can a quadratic formula calculator solve imaginary numbers?

How do you derive the quadratic formula with steps?

When should you solve a quadratic by factoring vs the quadratic formula?

Why is −b positive if b is negative?