Quadratic Formula Calculator (2024)

Using a Quadratic Formula Calculator

This calculator is an easy-to-use tool that solves quadratic equations. In algebra, a quadratic equation is any equation that can be written in the following form:

ax²+bx+c=0

where

a≠0

To use the quadratic formula calculator, enter the values of A, B, and C into the corresponding fields and press "Calculate." The value of A cannot equal zero, while zero is an acceptable input for B and C. For real and complex roots, the calculator will utilize the quadratic formula to determine all solutions to a given equation. After using the quadratic formula, the calculator will also simplify the resulting radical to find the solutions in their simplest form.

Solving quadratic equations using the quadratic formula

You can solve any quadratic equation with the quadratic formula. To use the quadratic formula, you should first bring the given equation to the following form: ax²+bx+c=0. Then, the solutions can be found as follows:

$$x=\frac{-b±\sqrt{b²-4ac}}{2a}$$

The part of the equation under the square root, b²-4ac, is called the discriminant.

  • If the discriminant is positive, b²-4ac>0, the equation will have two real roots.
  • If the discriminant is negative, b²-4ac<0, the equation will have two complex roots since the square root of a negative number is a complex number.
  • If the discriminant equals zero, b²-4ac=0, the equation will have only one root.

The quadratic equation calculator will display the solutions of the entered equations and the workflow of finding these solutions. The calculator will also calculate the discriminant and demonstrate whether it is positive, negative, or equal to zero.

Practical examples

Example 1 (with real roots)

Let's solve the quadratic equation:

2x²+3x-2=0

In this example

a=2,b=3,c=-2.

Using the quadratic formula for these values, we get:

$$x=\frac{-b±\sqrt{b²-4ac}}{2a}=\frac{-3±\sqrt{3^2-4(2)(-2)}}{2(2)}=\frac{-3±\sqrt{9--16}}{4}=\frac{-3±\sqrt{25}}{4}$$

The discriminant of this equation is positive,

b²-4ac=25>0

Therefore, the equation will have two real roots.

Now let's simplify the resulting radical:

$$x=\frac{-3±\sqrt{25}}{4}=\frac{-3±5}{4}$$

$$x=\frac{-3+5}{4}\ \ \ and\ \ \ x= \frac{-3-5}{4}$$

$$x=\frac{2}{4}\ \ \ and\ \ \ x=-\frac{8}{4}$$

$$x=\frac{1}{2}\ \ \ and\ \ \ x=-2$$

Finally

x=0.5

x=-2

Example 2 (with complex roots)

Let's solve the following quadratic equation:

x²+2x+5=0

In this example

a=1,b=2,c=5

Using the quadratic formula for these values, we get:

$$x=\frac{-b±\sqrt{b²-4ac}}{2a}=\frac{-2±\sqrt{2^2-4(1)(5)}}{2(1)}=\frac{-2±\sqrt{4-20}}{2}=\frac{-2±\sqrt{-16}}{2}$$

The discriminant of this equation is negative,

b²-4ac=-16<0

Therefore, the equation will have two complex roots.

Now let's simplify the resulting radical:

$$x=\frac{-2±\sqrt{-16}}{2}=\frac{-2±4i}{2}=\frac{-2}{2}±\frac{4i}{2}=-1±2i$$

Finally,

x=-1+2i

x=-1-2i

Example 3 (with one root)

Let's solve the following quadratic equation:

3x²+6x+3=0

In this example

a=3,b=6,c=3

Using the quadratic formula for these values, we get:

$$x=\frac{-b±\sqrt{b²-4ac}}{2a}=\frac{-6±\sqrt{6^2-4(3)(3)}}{2(3)}=\frac{-6±\sqrt{36-36}}{6}=\frac{-6±\sqrt0}{6}$$

The discriminant of this equation equals zero, b²-4ac=0. Therefore, the equation will have one root.

$$x=\frac{-6}{6}$$

Finally,

x=-1

Derivation of the quadratic formula

As demonstrated above, you can use the quadratic formula to solve absolutely any quadratic equation, regardless of whether the discriminant is positive, negative, or equals zero. Now let's investigate how it can be derived. Knowing the basic principles of formula derivation can be very useful in case you forget the formula itself.

The algorithm of quadratic formula derivation is relatively straightforward and is based on the procedure of completing the square. To derive the solutions of the standard quadratic equation ax²+bx+c=0, you need to follow the steps below:

  1. We have an equation:

ax²+bx+c=0

Move the constant C to the right side of the equation:

ax²+bx=-c

  1. Get rid of the coefficient A next to the squared term . To do this, divide the equation by A:

$$x²+\frac{b}{a}x=-\frac{c}{a}$$

  1. Add

$$(\frac{b}{2a})^2$$

to both sides of the equation:

$$x²+\frac{b}{a}x+(\frac{b}{2a})^2=-\frac{c}{a}+(\frac{b}{2a})^2$$

  1. The left-hand side now has the form

x²+2dx+d²

This expression can be rewritten as

(x+d)²

In our equation, d is expressed as

$$\frac{b}{2a}$$

So:

$$x²+\frac{b}{a}x+(\frac{b}{2a})^2 = \left(x+\frac{b}{2a}\right)^2$$

Substitute this into the left-hand side of our formula, and leave the right-hand side untouched for now:

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

Now the root x appears only once in the equation.

  1. Extract the square root from both parts of the equation:

$$x+\frac{b}{2a}=± \sqrt{-\frac{c}{a}+(\frac{b}{2a})^2}$$

  1. Move \$\frac{b}{2a}\$ to the right side of the equation:

$$x=-\frac{b}{2a}± \sqrt{-\frac{c}{a}+(\frac{b}{2a})^2}$$

  1. Multiply the right side of the equation by

$$\frac{2a}{2a}$$

$$x=\frac{-b ± \sqrt{-\frac{c}{a} × (2a)^2 + (\frac{b}{2a})^2 × (2a)^2}}{2a}$$

  1. Simplify the equation:

$$x=\frac{-b±\sqrt{-4ac+b²}}{2a}$$

  1. As a result, we get a quadratic formula:

$$x=\frac{-b±\sqrt{b²-4ac}}{2a}$$

Interesting facts about quadratic equation

  • The sum of the two roots of the quadratic equation is

$$\frac{-b}{a}$$

Consequently, if the discriminant of the quadratic equation b²-4ac equals zero, you can find the only root of the equation as

$$\frac{-b}{2a}$$

  • The product of the two roots of the quadratic equation is

$$\frac{c}{a}$$

  • The term "quadratic" comes from the Latin word "quadratus", which means "square." The equation was called quadratic since the highest power of the variable is 2, i.e., the variable is "squared."

  • The quadratic formula in its present shape was described as early as 628 AD by the Indian mathematician Brahmagupta, who didn't use symbols but instead discussed the solution using words. Brahmagupta, however, described only one of the two possible solutions, omitting the important ± sign before the square root.

  • The graph of a quadratic function y=ax²+bx+c is a parabola. The solutions, or roots, of the quadratic equation, are actually the coordinates of the interceptions of the graph with the x-axis. If the equation has two real roots, the graph intersects the x-axis twice. If the equation has only one root, the graph of the corresponding parabola only touches the x-axis at its maximum or minimum. If the equation has no real roots, the graph of the corresponding parabola does not intersect the x-axis at all.

  • When the value of the coefficient by the squared term, A, approaches zero, the graph of the corresponding parabola becomes flatter, eventually tending to become a straight line. When a=0, the equation becomes linear, and the graphical representation of it is obviously a straight line!

  • Similarly, when a>0, the parabola will be facing upwards. If a<0, the corresponding parabola will be opening downwards. If a=0, the "parabola" is flat, i.e., it is a straight line.

Quadratic equations are widely used in all areas of science. For example, in physics, quadratic equations are used to describe projectile motion.

Quadratic Formula Calculator (2024)

FAQs

How many answers should a quadratic equation have? ›

The quadratic equation always has two solutions of the variables. The solutions can be negative, positive, same, or different.

Does the quadratic formula always give 2 answers? ›

To solve a quadratic equation it must equal 0. A quadratic equation can have zero, one or two (real) solutions.

How do you know how many solutions a quadratic formula has? ›

The discriminant can be positive, zero, or negative, and this determines how many solutions there are to the given quadratic equation.
  • A positive discriminant indicates that the quadratic has two distinct real number solutions.
  • A discriminant of zero indicates that the quadratic has a repeated real number solution.

Does the quadratic formula give 2 answers? ›

If you are referring to the +/- that is before the square root, you end up doing both. Quadratic equations will have 2 solutions unless the square root = 0, then it degrades to just one solution. The +/- in front of the square root is what creates the two different solutions.

Can a quadratic have one answer? ›

Answer: If a quadratic equation has exactly one real number solution, then the value of its discriminant is always zero. A quadratic equation in variable x is of the form ax2 + bx + c = 0, where a ≠ 0.

Does every quadratic equation have 2 answers? ›

As we have seen, there can be 0, 1, or 2 solutions to a quadratic equation, depending on whether the expression inside the square root sign, (b2 - 4ac), is positive, negative, or zero. This expression has a special name: the discriminant.

Does the quadratic formula ever not work? ›

Solve x2 + 3x − 4 = 0

So, as expected, the solution is x = −4, x = 1. For this particular quadratic equation, factoring would probably be the faster method. But the Quadratic Formula is a plug-n-chug method that will always work.

Does the quadratic formula work every time? ›

Finally, the quadratic formula will work on any quadratic equation. However, if using the formula results in awkwardly large numbers under the radical sign, another method of solving may be a better choice.

How to find out how many solutions an equation has? ›

If we can solve the equation and get something like x=b where b is a specific number, then we have one solution. If we end up with a statement that's always false, like 3=5, then there's no solution. If we end up with a statement that's always true, like 5=5, then there are infinite solutions.. Created by Sal Khan.

Can a quadratic equation have no solution? ›

These values are called the solutions of the equation. Linear equations that are written in the standard form , ax + b = 0, a ≠ 0, have one solution. Quadratic equations may have no solutions, one solution, or, as in the above example, two solutions.

How many solutions will we usually get when we apply the quadratic formula? ›

Solving quadratic equations

If this is negative, there are no real solutions to the equation. If the discriminant is zero, there is only one real solution. If the discriminant is positive, then the symbol indicates that you will get two real solutions.

Can Photomath solve quadratic equations? ›

In our opinion, this is where math learning really gets exciting! We've got loads of algebra coverage so that you can stay calm and collected, even when letters start showing up. Quadratic equations, linear equations, inequalities… Chances are, if you're solving for a variable, we can help you find it.

Can you use the quadratic formula without a calculator? ›

You factorise it. First, you find a common factor between a, b and c. Then, you find the 2 numbers that multiplied give you c and added give you b.

Why does my calculator say math error when I do a quadratic formula? ›

It may mean that your equation has no solutions: for instance, if you are trying to find out where two curves meet, a Math Error would indicate that they don't meet anywhere. You will also get a Math Error if you try to take the logarithm of zero or a negative number (why is this?).

How many solutions could a quadratic function have? ›

A quadratic equation has two solutions. To find out what type of solutions you get, you need to find the value of the discriminant. For a quadratic equation ax2+bx+c=0 a x 2 + b x + c = 0 , the discriminant is b2−4ac b 2 − 4 a c , which is also the value inside the square root in the quadratic formula.

How many points do you need to solve a quadratic equation? ›

A quadratic function, of the form f(x) = ax2 + bx + c, is determined by three points. Given three points on the graph of a quadratic function, we can work out the function by finding a, b and c algebraically. This will require solving a system of three equations in three unknowns.

What is the maximum number of answers solutions that a quadratic can have? ›

A quadratic equation can have at most two solutions (and actually always has exactly two complex solutions - a fact you may learn later in your studies).

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