Quadratic Formula Calculator – Free Solver

A quadratic formula calculator is a helpful tool that can be used to solve quadratic equations quickly and efficiently. This calculator is perfect for high school students, college students, teachers and those who use algebra to solve problems.

A quadratic equation is a second order polynomial equation. Generally a quadratic equation will contain a variable that is squared. The standard form of a quadratic equation is:

$$

ax^2+bx+c=0

$$

Where a, b and c are numbers and x is the unknown variable. a should not be equal to zero.

To solve a quadratic equation we are trying to find the value(s) of x that make the equation equal to zero. These values are known as solutions or roots.

This quadratic formula calculator uses the standard quadratic formula to find the solutions. The calculator will also tell you the discriminant and whether the roots are real, repeating or complex.

By using this calculator you can quickly find the solution to your equation by entering in the coefficients. This calculator can be used to help with algebra homework, in class and revision.

How to use the quadratic formula calculator

To use this quadratic formula calculator simply follow the steps below.

• Enter Coefficient A: Enter in the number that is multiplied by the squared variable (x2)
• Enter Coefficient B: Enter in the number that is multiplied by the variable (x)
• Enter Coefficient C: Enter in the constant number.
• Double check your values: Make sure the coefficients match your original quadratic equation.
• Click calculate: Click the calculate button to find the solution.
• View your solutions: View the roots, discriminant and type of solution.

For example if your equation is x2-5x+6=0. Enter in:

a0__ = -a1__

The calculator will give you two real solutions of a2__ and a3__.

Quadratic Formula Calculator Formula

The quadratic formula is a way of solving any quadratic equations that can be written in the standard form.

$$

a4__

$$

$$

a5__

$$

$$

a6__

$$

Positive discriminant = 2 real roots.

$$

a7__

$$

$$

a8__

$$

$$

a9__

$$

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Where:

a = Coefficient of the squared term (b0__)

b = Coefficient of the linear term (b1__)

c = Constant term.

x = Unknown variable/solution.

Discriminant

The part of the equation inside the square root is known as the discriminant.

$$

b2__-4ac

$$

This tells us the type and amount of solutions we have.

If D is positive – There are two distinct real solutions.

If D is equal to zero – There is one repeating real solution.

If D is negative – There are two complex conjugate solutions.

The discriminant is important as it tells us the type of solution we have.

Example

Let’s say we want to solve the following quadratic equation.

$$

x^2-5x+b3__

$$

Our coefficients are:

Coefficient Value

a 1

b -5

c 6

Step 1 – Find the Coefficients

Compare the equation to the standard quadratic equation.

$$

b4__

$$

Therefore,

b5__ = -b6__

Step 2 – Calculate the discriminant

$$

b7__-4ac

$$

$$

D=(-5)^2-4(1)(6)

$$

$$

b8__

$$

As the discriminant is positive, there are two distinct real solutions.

Step 3 – Apply the quadratic formula.

$$

x=\\frac{-(-5)\\pm\\sqrt{1}}{2(1)}

$$

$$

x=\\frac{5\\pm1}{2}

$$

Step 4 – Calculate Both Roots

First solution:

$$

x_1=\\frac{5+1}{2}=3

$$

Second solution:

$$

x_2=\\frac{5-1}{2}=2

$$

Solutions = b9__ and c0__

So our two roots are 2 and 3.

Understanding the Results

After entering in your coefficients you will be given your solutions along with other useful maths information.

First root (x₁): This is the first solution to your equation.

Second root (x₂): This is the second solution to your equation.

Discriminant: This is the value of c1__. This tells us what type of solution we have.

Type of solution: Tells you whether there are two distinct real roots, one repeating real root or two complex roots.

Vertex: This is the turning point of your quadratic function.

Axis of symmetry: This is the vertical line that goes through the vertex of the parabola.

For example if your calculator gives you c2__ and c3__, both of these values will fit the original equation.

You can plug these values back into the original equation to check your answers.

Factors that effect the result

There are a few factors that will affect the solution of the quadratic equation.

Coefficient A: This tells us what the squared term is. This cannot be zero as the equation won’t be quadratic anymore.

Coefficient B: This tells us the position of the roots and axis of symmetry.

Coefficient C: This tells us where on the y-axis the quadratic graph will be.

Discriminant: This tells us what type of roots we have, either two distinct real roots, one repeating real root or complex roots.

Positive/Negative Coefficients: Depending on what combination of positive and negative coefficients you have can affect your solution.

Complex Numbers: If the discriminant is negative then your solution will include imaginary numbers. This is represented by the imaginary unit c4__. Where c5__.

Wrong values entered: If you enter in the wrong sign, miss out a decimal place or enter in the wrong coefficient you will get incorrect results.

Benefits and uses of a quadratic formula calculator

A quadratic formula calculator has many benefits. It can help students, teachers and professionals.

• Save time: Quickly solve your quadratic equations.
• More accurate: Reduce common errors made when applying the quadratic formula.
• Understand algebra better: Understand how the coefficients, discriminant and roots all relate.
• Check homework: Check your answers to algebra homework and practice.
• Revise for exams: Practice solving different types of quadratic equations.
• Understand complex roots: Know which equations have imaginary/complex solutions.
• Help with graphing: Use the roots to understand where the parabola will cross the x-axis.

Quadratic equations can be used in real life situations such as physics, engineering, economics and construction. They can be used to model projectile motion, calculate area and analyse relationships involving changing quantities.

Accuracy, assumptions and limitations

The quadratic formula calculator uses a standard formula to find the roots of any quadratic equation. If you enter in valid values then you will get accurate results however there are some limitations to the calculator due to the computer’s precision.

Assumptions and limitations:

Equation must be in the form c6__.

Coefficient c7__ cannot be zero.

All three coefficients must be valid.

Decimal and negative coefficients are allowed.

Negative discriminant will give complex solutions.

Large or small coefficients could cause floating point errors.

Decimal answers may be rounded.

To get the most accurate results enter your coefficients carefully and double check by plugging your solutions back into the equation.

This calculator is for educational purposes only.

FAQs

1. What is a quadratic formula calculator?

A quadratic formula calculator is an online maths tool that can be used to solve any quadratic equations using the standard quadratic formula. The calculator will use the coefficients to find the roots of the equation. The calculator can also tell you whether the solutions are real, repeating or complex.

2. What is the standard form of a quadratic equation?

A quadratic equation is in the form c8__. Where a, b and c are numbers. a cannot be zero. x is the unknown value you are trying to find.

3. Can there be two solutions to a quadratic equation?

Yes! There can be two distinct real solutions to a quadratic equation. This will happen if the discriminant is positive. However, if the discriminant is zero there will be one repeating real solution. If the discriminant is negative there will be two complex solutions.

4. What is the discriminant?

The discriminant is the part of the equation c9__. This is found inside the square root. This tells us what type of roots we have. If the discriminant is positive there will be two distinct real roots. If the discriminant is zero there will be one repeating real root. If the discriminant is negative there will be two complex roots.

5. What happens if the discriminant is negative?

If the discriminant is negative there will not be any real number solutions. There will be two complex conjugate solutions. The solutions will include the imaginary unit i. Where x0__.

6. Can I use decimals and negatives?

Yes! The quadratic formula can be used with positive numbers, negative numbers, integers and decimal coefficients. Just make sure you enter in the correct signs and values for a, b and c to get accurate results.

7. How do I check my solutions?

You can plug your solutions back into the original quadratic equation. If the equation equals zero then your solution is correct. For example if you plug in x1__ into x2__ you get x3__.