Quadratic Formula Calculator With Step-by-Step Solutions

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Use a particular theorem?

What is a quadratic function?

A quadratic function is a polynomial of degree two, written as

f(x)=ax^2+bx+c

, where

a≠0

. Its graph is a parabola that opens up if

a>0

or down if

a<0

Generic parabola: f(x)=x²

What is a quadratic equation?

A quadratic equation sets the function equal to zero: $ax^2+bx+c=0$ . Its solutions give the x-intercepts (roots) of the parabola.

Example parabola: f(x)=2x²−4x−6 with roots at x=−1,3

What is the quadratic formula?

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

This solves $ax^2+bx+c=0$ in terms of $a$ , $b$ , and $c$ .

How they fit together

The quadratic function $f(x)=ax^2+bx+c$ leads to the quadratic equation $f(x)=0$ . The quadratic formula then solves that equation, yielding the parabola’s roots, which correspond to its x-intercepts.

How to solve a quadratic equation step by step

Step 1: Identify coefficients $a$ , $b$ , and $c$ from $ax^2 + bx + c = 0$ .

Step 2: Compute the discriminant $D = b^2 - 4ac$ .

Step 3: Plug into the formula $x = \frac{-b \pm \sqrt{D}}{2a}$ .

Step 4: Determine root types: two real roots if $D > 0$ , one repeated root if $D = 0$ , or complex if $D < 0$ .

Example: Solve $2x^2 - 4x - 6 = 0$

a = 2,\; b = -4,\; c = -6 \\ D = (-4)^2 - 4\cdot2\cdot(-6) = 16 + 48 = 64 \\ x = \frac{-(-4)\pm\sqrt{64}}{2\cdot2} = \frac{4\pm8}{4} \;\Rightarrow\; x = 3,\;-1

The parabola opens up (since $a = 2 > 0$ ), with roots at $x = 3$ and $x = -1$ .

Tips & Tricks

If $b^2 - 4ac$ is negative, expect complex roots—express them as $p \pm qi$ .
Use the vertex $\bigl(-\tfrac{b}{2a},\,f(-\tfrac{b}{2a})\bigr)$ to sketch the parabola quickly.
For integer solutions, check if the discriminant is a perfect square to simplify your work.