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Education Calculators / Quadratic Formula Calculator

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Examples

Exp: 1 | 3x^{2} - 6x + 12 = 0 |

Exp: 2 | x^{2} + 3x - 8 = 0 |

Exp: 3 | 7x^{2} + 11x + 7 = 0 |

Exp: 4 | x^{2} - 4x + 4 = 0 |

Exp: 5 | 2.5x^{2} - 1.9x + 5 = 0 |

This quadratic formula calculator is work as a quadratic equation solver that helps to solve a given quadratic equation by using the quadratic equation formula.

Well, before knowing about this quadratic equation calculator, let’s start with some basics!

The quadratic formula is said to be one of the most potent tools in mathematics. This formula is the solution of a second-degree polynomial equation. The standard form of a quadratic equation is mentioned-below:

**ax ^{1} + bx + c = 0**

Where;

- ‘a’ is the quadratic coefficient
- ‘x’ is the unknown
- ‘b’ is the linear coefficient
- ‘c’ is the constant

The solution of this equation is said to be as the root of the equation.

Well, a quadratic equation has at most two roots, so solving quadratic equations ultimately means finding the roots of a quadratic equation. However, at first, complex equations are get simplified to make it in standard form. Thus, the values of ‘a’, ‘b’, and ‘c’ are used in the quadratic formula equation to find the roots.

The given quadratic formula for finding the roots is:

**\[ x = \dfrac{ -b \pm \sqrt{b^2 – 4ac}}{ 2a } \]**

In order to analyze the nature of the solution; the discriminant is figured out as:

D = b^{2} – 4ac

The b^{2} – 4ac is said to be as Discriminant. These two roots are calculated once by putting the positive sign and another one by putting a negative sign.

**\[ x₁ = \dfrac{ -b + \sqrt{b^2 – 4ac}}{ 2a } \]**

**\[ x₂ = \dfrac{ -b – \sqrt{b^2 – 4ac}}{ 2a } \]**

Our quadratic formula calculator also uses the same formula to solve quadratic equation.

There are three possibilities of getting the roots of quadratic equation, but remember that these possibilities depend upon the value of Discriminant.

- If b
^{2}– 4ac = 0, then there will be only one root - If b
^{2}– 4ac > 0, then there will only two real roots - If b
^{2}– 4ac < 0, then there will be two complex roots

Coefficients of a quadratic equation:

Also, it is important to note that the numerals i.e. a, b, and c are said to be the coefficient of the equation and they cannot be ‘0’. They all are real numbers that not dependent on x. If A = 0, then the equation is not said to quadratic, but linear.

If B² < 4AC, then the determinant Δ will be negative, it is said to be as this an equation has no real roots.

Our quadratic calculator can also help you if you can put the equation in this form:

**ax ^{2} + bx + c = 0**

This quadratic formula calculator is a tool that helps to solve a quadratic equation by using a quadratic formula or complete the square method. You just have to form of an equation, computation method, and type the parameters of the equation; this quadratic formula solver will work best for you!

Don’t fret; this quadratic equation solver is quite easy to use and loaded with smart and user-friendly interface!

You have to select the form of equation; this is the form according to which you have to enter the values into the designated fields of our quadratic function calculator.

This calculator uses the following form:

- Ax
^{2}+ Bx + C=0 (Standard Form) - A(x – H)
^{2}+ K =0 (Vertex Form) - A(x-x₁)(x-x₂)= 0 (Factored Form)

Our quadratic equation calculator allows you to solve the quadratic equation by using the quadratic formula and completing the square method

If you selected Ax^{2} + Bx + C=0 form, then you have to enter the values of A, B, and C

If you selected A(x – H)^{2} + K =0 form, then you have to enter the values of A, H, and K

If you selected A(x-x₁)(x-x₂)= 0 form, then you have to enter the values of A, x1, and x2

Once you entered the above values, then our quadratic equation solver shows the following:

This quadratic root calculator shows the root or roots of your given equation.

The calculator simplify the given equation step-by-step.

If you solve the quadratic equation by using the quadratic formula, then our quadratic discriminant calculator show the discriminant

This quadratic graph calculator shows you the complete quadratic graph for a given equation!

When it comes to solving quadratic equations, quadratic formula is account to perform calculations. So, it is important to learn it by heart, not only how to derive it, but also how to make use of it.

The standard form of a quadratic equation is as follow:

**ax ^{2} + bx + c = 0**

with a ≠ 0, it has the solution of the form:

**\[ x = \dfrac{ -b \pm \sqrt{b^2 – 4ac}}{ 2a } \]**

And, the discriminant is defined as;

D = b^{2} – 4ac

There are different quadratic formula steps that you have to follow to get successful quadratic equation solution:

First, of all examine the give of the form of ax^{2} + bx + c, and then determine the coefficients a, b, and c. The ‘a’ is said to be coefficient that appears multiplying the quadratic term x^2x. The ‘b’ is said to be coefficient that appears multiplying the linear term x, and the ‘c’ coefficient is said to be constant.

What are the coefficients of the following expression x^{2} + 3x + 1?

In this case a=1 (it is the coefficient that multiplying by the quadratic term x^{2}), b=3b=3 (the coefficient that multiplying by the linear term x), and c=1 (the constant).

What are the coefficients now, if you have the following expression: 5/4 + 3/4 x+1/2 x^{2}

In this case a=1/2 (it is the coefficient that multiplying by the quadratic term x^{2}), b=3/4 (the coefficient that multiplying by the linear term x), and c=5/4 (the constant).

What are the coefficients, if you have the following expression: -3 + 1/2

In this case, a=0 as the given expression doesn’t contain a quadratic term x^{2}. So, this is not said to be a quadratic expression.

The formula is:

\[ x = \dfrac{ -b \pm \sqrt{b^2 – 4ac}}{ 2a } \]

Now, you have to replace the value of the coefficients a, b, and c.

If the given equation is -3x^{2} + 2x – 1 = 0 coefficients, from above examples you will know that what are the coefficients in this expression. Here, a=-3, b=2, and c=1

So, by plugging the values into the formula we get:

\[ x = \dfrac{ -2 \pm \sqrt{2^2 – 4(-3)(1)}}{ 2(-3) } \]

Once you plugged the values of a, b, and c, you have to simplify the values in the equation. From the previous example, you have:

\[ x = \dfrac{ -2 \pm \sqrt{4 – 12}}{(-6) } \]

\[ x = \dfrac{ -2 \pm \sqrt{-8}}{(-6) } \]

If the value is positive, then equation has two real roots. If the value is 0, then there is only one real root, and if the value inside of the square root is negative, then there will be two complex roots. In the previous example, you have a -8 inside of the square root, means you have two complex solutions (as shown below):

\[ x = \dfrac{ -2 \pm \sqrt{4 – 12}}{(-6) } \]

\[ x = \dfrac{ -2 \pm \sqrt{-8}}{(-6) } \]

\[ x = \dfrac{ -2 \pm \sqrt{8} \,i}{(-6) } \]

Thankfully, you come to know how to solve quadratic equations (manually). But, what if you don’t want to stick to these given quadratic formula steps, don’t fret! Get the quadratic equation solution by using our quadratic formula calculator with steps calculations!

Yes, our quadratic formula solver indicates when the equation has no real roots, it helps to find the solution of a quadratic equation with a negative determinant. These roots will be said to as complex numbers.

Complex numbers have a real and imaginary part, remember that the imaginary part is always equal to the number i = √(-1) multiplied by a read number.

In fact, the formula for the quadratic expression remains the same in this case:

\[ x = \dfrac{ -b \pm \sqrt{b^2 – 4ac}}{ 2a } \]

Keep in mind, as b2 – 4ac < 0, the square root of the determinant will be an imaginary value. Hence:

Re(x) = -B/2A

Im(x) = ± (√Δ)/2A

Well, from the graph of the parabola find out the vertex, axis of symmetry, y-intercept, x-intercept.

The problem has two solutions and they demonstrate the intersecting points of the equation, which is the x-intercept (it is the point where the x-axis is crisscrossed by a curve. Whilst, preparing a graph of the given equation x^{2} + 3x – 4 = 0, can be viewed as:

It demonstrate peak. So, the vertex of quadratic equation indicates the peak point of the parabola. If the parabola opens upward, it said to be the vertex is the highest point, and if the parabola opens downward, so the vertex is said to be the lowest point.

The axis of symmetry divides the parabola into two equal halves; it always passes through the vertex of parabola.

Roots are also referred as the x-intercept. It is allocated below the x-axis or above the x-axis, in the graph. That’s why, to determine the root of a quadratic function, we set y=0

Every parabola has y-intercept, it is said to be the point at which the function crosses the y-axis. It is figure out by setting the x-variable in the equation to 0.

So, let’s start solve graphically,

First, take the equation f (x) = 2x^{2} – 4x-1 or Y = 2x^{2} – 4x-1

Here, a = 2, b = -4, and c = -1

If ‘a’ having the positive value, then remember that the parabola opens upward in graph. First, you have to find the vertex of x:

x=(-b)/2a

x=(-(-4))/2(2)

x=1

Now, you have to find the vertex of Y:

You have to plug the value of x in equation 2x^{2} – 4x-1

y = 2(1)^{2} – 4(1)-1

y= 2 – 4 – 1

y= 3

So, you have axis of symmetry: x = 1

Now, you have to find the x-intercept by using quadratic formula:

\[ x = \dfrac{ -(-4) \pm \sqrt{(-4)^2 – 4(2)(-1)}}{2(2) } \]

\[ x = \dfrac{ 4 \pm \sqrt{16 + 8}}{ 4 } \]

\[ x = \dfrac{ 4 \pm \sqrt{24}}{ 4 } \]

\[ x = \dfrac{ 4 \pm 4.9}{ 4 } \]

\[ x = \dfrac{ 4 + 4.9}{ 4 } , x = \dfrac{ 4 – 4.9}{ 4 } \]

X-intercept = 2.23, – 0.023

Now, you have to find y-intercept, put the value of x=0 in equation as:

y= 2x^{2} – 4x – 1

y= 2(0)^{2} – 4(0) -1

y-intercept = -1

Now, let us plot the values into the graph:

Graph:

Now, let take another equation in which parabola opens downward.

-x^{2} + 2x + 1 = 0

If ‘a’ containing negative value, then the parabola opens downward

Now, find vertex of x:

x=(-b)/2a

x=(-2)/2(-1) =1

Fid vertex of y:

Now, you have to put the value of x in equation,

Y = -(1)^{2} + 2(1) + 1

Y =2

Now, find x-intercept by using quadratic equation:

\[ x = \dfrac{ -b \pm \sqrt{b^2 – 4ac}}{ 2a } \]

a = -1, b = 2, c = 1;

\[ x = \dfrac{ -2 \pm \sqrt{2^2 – 4(-1)(1)}}{ 2(-1) } \]

\[ x = \dfrac{ -2 \pm \sqrt{8}}{ -2 } \]

x1 = − 0.414214

x2 = 2.414214

Now, find y-intercept:

x^{2} + 2x + 1 = 0

(0)^{2} + 2(0) + 1 = 0

y-intercept = 1, now you have to plot the values into the graph!

The quadratic formula is a well-known formula that found everywhere in mathematics. It often accounts when you are solving all kind of geometric problems such as:

- Maximizing An Area
- Given a Fixed Perimeter
- Numerous Word problems

There are lots of individuals who wonder if there is any relationship between this quadratic equation formula and the method of completing the square. In simple term, you get the quadratic formula by simply solving the quadratic equation via completing the square. It is the same idea exactly, which derives to the quadratic formula that we all know!

As a student, you might be taken into account on various regarding mathematics. Also, students generally make use of this equation in subjects like engineering and physics. There are some other professions that use quadratic equations:

- Military and Law Enforcement – (to find the trajectory of missiles fired by artillery)
- Engineers – (Relates to Civil Engineering)
- Equation in motion (playground as well and in gaming situations, it describing the trajectory of a ball and determining the height of a thrown ball)
- Science (Astronomers – describe the orbit of planets,solar systems and galaxies)
- Agriculture sectors (optimal arrangement of boundaries to produce the biggest field)

- Simply, you just have to complete the square of ax
^{2}+ bx + c = 0 to get the quadratic formula - You ought to divide both sides of the equation by ‘a’, so the coefficient of x
^{2}is 1 - So, you ought to rewrite the left side is in the form of x^2+ bx (although in this case, bx is actually

- You have to put all the terms on one side of the equal sign, leaving zero on the other side
- Factor
- Then, you ought to set each factor equal to zero
- Very next, you have to solve each of these equations
- Finally, you have to check by inserting your answer in the original equation

If the quadratic equation ax^{2} + bx + c = 0, has no ‘b’ term, then, it means it has the form 〖ax〗^2+ c=0. In such case, you can solve this equation by using the simple square root property.

It helps to determine how many solutions there are to the quadratic equation. If the discriminant is positive, then it is said that there are 2 roots. If it is zero, then there is only 1 root. If the discriminant is negative, then it is said that there are 0 roots.

Solve Any Quadratic Equation – Understanding The Quadratic Formula

By PurpleMath – Solve Each Quadratic Equation “The Quadratic Formula Explained”

By MathIsFun – Algebra Index – Quadratic Equation Solver

By CliffNotes – Study Guide (Algebra) – Solving Quadratic Equations