## $ax^2 + bx +c = 0$

### x2 + x + = 0

$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$

Quadratic Formula Calculator is a powerful webapplication that allows you to solve complex second-order polynomial equation such as ax2 + bx + c = 0 for x, where a ≠ 0.

The "b2-4ac" part of the Formula is called the “discriminant”, This part of the Formula produces different types of solutions.

The calculator determines whether the discriminant (b2−4ac) is less than, greater than, or equal to 0.

When $b^2 -4ac = 0$ there is one real root.
When $b^2 -4ac> 0$ there are two real roots.
When $b^2 -4ac < 0$ there are two complex roots.

#### Solution

find the solution of the given equation

#### $x^2$$x$ $= 0$

using the Quadratic Formula where $a =$ , $b =$ , and $c =$

$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}$$
x = - (0) ± (0)2 - 4 .(0) . (0) 2 . (0)
x = - (0) ± (0)2 - (0) 2 . (0)
x = - (0) ± (0) - (0) 2 . (0)
x = - (0) ± 0 2 . (0)
x = - (0) ± 0 0

The discriminant $b^2 -4ac> 0$ so, there are two real roots.

x = 0 ± 0 0
x = 0 + 0 0          AND          x = 0 - 0 0
x = 0 0          AND          x = 0 0

which becomes

x = 0          AND          x = 0

The discriminant b2−4ac=0 so, there is one real root.

x = 0

The discriminant $b^2 -4ac> 0$ so, there are two real roots.

The discriminant $b^2 -4ac < 0$ so, there are two complex roots.

x = ±
x = ( ±)
x = ( ±)

which becomes:

x = +          AND          x = -

which becomes:

x = +          AND          x = -
x = 0          AND          x = 0

## More Examples

#### Example 1:

Find the Solution for $2x^2 +4x +2 = 0$, where $a = 2$, $b = 4$ and $c = 2$, using the Quadratic Formula.
$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
x = - (4) ± (4)2 - 4 .(2) . (2) 2 . (2)
x = - (4) ± (4)2 - (16) 2 . (2)
x = - (4) ± (16) - (16) 2 . (2)
x = - (4) ± 0 2 . (2)

The discriminant $b^2 -4ac = 0$ so, there is one real root.

x = -1

#### Example 2:

Find the Solution for $x^2 -3x +10 = 0$, where $a = 1$, $b = -3$ and $c = 10$, using the Quadratic Formula.
$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
x = - (-3) ± (-3)2 - 4 .(1) . (10) 2 . (1)
x = - (-3) ± (-3)2 - (40) 2 . (1)
x = - (-3) ± (9) - (40) 2 . (1)
x = - (-3) ± -31 2 . (1)

The discriminant $b^2 -4ac < 0$ so, there are two complex roots.

x = 3 ±31i 2

which becomes

x = 3 + 31i 2          AND          x = 3 - 31i 2

#### Example 3:

Find the Solution for $3x^2 -5x + 6 = 0$, where $a = 3$, $b = -5$ and $c = 6$, using the Quadratic Formula.
$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
x = - (-5) ± (-5)2 - 4 .(3) . (6) 2 . (3)
x = - (-5) ± (-5)2 - (72) 2 . (3)
x = - (-5) ± (25) - (72) 2 . (3)
x = - (-5) ± -47 2 . (3)

The discriminant $b^2 -4ac < 0$ so, there are two complex roots.

x = 5 ±47i 6

which becomes:

x = 5 + 47i 6          AND          x = 5 - 47i 6

#### Example 4:

Find the Solution for $6x^2 +11x -35 = 0$, where $a = 6$, $b = 11$ and $c = -35$, using the Quadratic Formula.
$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
x = - (11) ± (11)2 - 4 .(6) . (-35) 2 . (6)
x = - (11) ± (11)2 - (-840) 2 . (6)
x = - (11) ± (121) - (-840) 2 . (6)
x = - (11) ± 961 2 . (6)
x = - (11) ± 31 12

The discriminant $b^2 -4ac > 0$ so, there are two real roots.

x = -11 ± 31 12
x = -11 + 31 12          AND          x = -11 - 31 12
x = 20 12          AND          x = -42 12

which becomes

x = 1.67          AND          x = -3.50

#### Example 5:

Find the Solution for $-x^2 +7x -10 = 0$, where $a = -1$, $b = 7$ and $c = -10$, using the Quadratic Formula.
$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
x = - (7) ± (7)2 - 4 .(-1) . (-10) 2 . (-1)
x = - (7) ± (7)2 - (40) 2 . (-1)
x = - (7) ± (49) - (40) 2 . (-1)
x = - (7) ± 9 2 . (-1)
x = - (7) ± 3 -2

The discriminant $b^2 -4ac > 0$ so, there are two real roots.

x = -7 ± 3 -2
x = -7 + 3 -2          AND          x = -7 - 3 -2
x = -4 -2          AND          x = -10 -2

which becomes

x = 2          AND          x = 5

#### Example 6:

Find the Solution for $10x^2 -4x + 10 = 0$, where $a = 10$, $b = -4$ and $c = 10$, using the Quadratic Formula.
$$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$
x = - (-4) ± (-4)2 - 4 .(10) . (10) 2 . (10)
x = - (-4) ± (-4)2 - (400) 2 . (10)
x = - (-4) ± (16) - (400) 2 . (10)
x = - (-4) ± -384 2 . (10)

The discriminant $b^2 -4ac < 0$ so, there are two complex roots.

x = 4 ±86i 20
x = 4 (1 ±26)i 20
x = (1 ±26)i 5

which becomes:

x = 1 + 26i 5          AND          x = 1 - 26i 5

Quadratic Formula Calculator is a powerful web application that allows you to solve complex second-order polynomial equations such as ax2 + bx + c = 0. Using the quadratic tool, you can solve such polynomial equations and find the value of x, so long as a ≠ 0. The quadratic tool will be a boon for those who need to solve complex polynomial equations for their work. It will save precious time and effort. Let’s take a look at how the tool works.

## How Quadratic Formula Calculator Works

The quadratic tool uses the Quadratic Formula, which is $x = {-b \pm \sqrt{b^2-4ac} \over 2a}$ to solve advanced and complex second-order polynomial equations. In the Formula, “±” is used to do both addition and subtraction operations. That’s because, usually, there are two solutions to a second-order polynomial equation.

The “b2-4ac” ppart of the Formula is called the“discriminant,” This part of the Formula produces different types of solutions. The “a,” “b,” and “c” in the Formula are numerical, where “a” cannot be equal to zero. The “x” is a variable that you will need to find. It’s also called the root of the equation.

Let’s look at some solutions you can get from a quadratic equation.

Solution 1: If the discriminant is positive, that is, there is a “+” instead of the “±” part, then we get two real solutions.

Solution 2: If the discriminant is zero, we get only one answer.

Solution 3:If there’s a negative solution, you get complex solutions.

## What Makes Quadratic Formula Calculator So Unique?

Here are some characteristics that make the Quadratic Formula calculator so unique.

• The quadratic tool uses a clean and straightforward user interface.
• Our tool working process is clearly explained.
• We use open-source technologies to design Quadratic tools.

## How To Use Quadratic Formula Calculator?

The quadratic tool uses the Quadratic Formula to solve any second-order polynomial equations that you have at hand. A quadratic equation calculator can be used as a verification tool to verify the answers to the equations you solve manually. Follow these steps to learn how to use the Quadratic tool.

Step 1: Enter the values coefficients of your quadratic equation, such as “a,” “b,” and “c” in the input fields of the tool.

Step 2: Below the input fields, there’s a button called “Calculate Quadratic Roots.” Click that button.

Step 3: Once you click that button, the result of the quadratic equation will be displayed on the screen in the form of the discriminant and the roots.

## How Are Quadratic Equations Represented in a Graph?

When a quadratic equation of second order is represented in a graph, it is called a parabola. Here’s a representation of a parabola. For this, let’s assume the equation is ax2 + bx + c = 0. The Formula that we use to draw the graph is $x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$