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Quadratic Equation Solver

We can help you solve an equation of the form " ax 2 + bx + c = 0 " Just enter the values of a, b and c below :

Is it Quadratic?

Only if it can be put in the form ax 2 + bx + c = 0 , and a is not zero .

The name comes from "quad" meaning square, as the variable is squared (in other words x 2 ).

These are all quadratic equations in disguise:

How Does this Work?

The solution(s) to a quadratic equation can be calculated using the Quadratic Formula :

The "±" means we need to do a plus AND a minus, so there are normally TWO solutions !

The blue part ( b 2 - 4ac ) is called the "discriminant", because it can "discriminate" between the possible types of answer:

• when it is positive, we get two real solutions,
• when it is zero we get just ONE solution,
• when it is negative we get complex solutions.

Learn more at Quadratic Equations

Quadratic Formula Calculator

What is the quadratic formula, coefficients of a quadratic equation, how to use the quadratic formula solver, solving quadratic equations with a negative determinant, extra resources.

If you need to solve an equation of the form Ax² + Bx + C = 0 , this quadratic formula calculator is here to help you. With just a few clicks, you will be able to solve even the most challenging problems. This article describes in detail what is the quadratic formula and what the symbols A, B, and C stand for. It also explains how to solve quadratic equations, which have a negative determinant and no real roots.

The quadratic formula is the solution of a second-degree polynomial equation of the following form:

Ax² + Bx + C = 0

If you can rewrite your equation in this form, it means that it can be solved with the quadratic formula. A solution to this equation is also called a root of an equation.

The quadratic formula is as follows:

x = (-B ± √Δ)/2A

• Δ = B² – 4AC

Using this formula, you can find the solutions to any quadratic equation. Note that there are three possible options for obtaining a result:

The quadratic equation has two unique roots when Δ > 0. Then, the first solution of the quadratic formula is x₁ = (-B + √Δ)/2A , and the second is x₂ = (-B – √Δ)/2A .

The quadratic equation has only one root when Δ = 0. The solution is equal to x = -B/2A . It is sometimes called a repeated or double root.

The quadratic equation has no real solutions for Δ < 0.

You can also graph the function y = Ax² + Bx + C . It's shape is a parabola, and the roots of the quadratic equation are the x-intercepts of this function.

💡 We use the quadratic formula in many fields of our life, not only mathematics or physics but also construction. For example, you can plan a smooth transition between two sloped roadways using the vertical curve formula which bases on the quadratic equation.

A, B, and C are the coefficients of the quadratic equation. They are all real numbers, not dependent on x. If A = 0, then the equation is not quadratic, but linear.

If B² < 4AC , then the determinant Δ will be negative. It means that such an equation has no real roots.

Write down your equation. Let's assume it is 4x² + 3x – 7 = -4 – x .

Bring the equation to the form Ax² + Bx + C = 0 . In this example, we will do it in the following steps:

4x² + 3x - 7 = -4 – x

4x² + (3 + 1)x + (-7 + 4) = 0

4x² + 4x - 3 = 0

Calculate the determinant.

Δ = B² – 4AC = 4² - 4×4×(-3) = 16 + 48 = 64 .

Decide whether the determinant is greater, equal, or lower than 0. In our case, the determinant is greater than 0, which means that this equation has two unique roots.

Calculate the two roots using the quadratic formula.

x₁ = (-B + √Δ)/2A = (-4 +√64) / (2×4) = (-4 + 8) / 8 = 4/8 = 0.5

x₂ = (-B – √Δ)/2A= (-4 -√64) / (2×4) = (-4 – 8) / 8 = -12/8 = -1.5

The roots of your equation are x₁ = 0.5 and x₂ = -1.5 .

You can also simply type the values of A, B, and C into our quadratic equation calculator and let it perform all calculations for you.

Make sure you have written down the correct number of digits using our significant figures calculator .

Even though the quadratic formula calculator indicates when the equation has no real roots, it is possible to find the solution of a quadratic equation with a negative determinant. These roots will be complex numbers .

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

The quadratic formula remains the same in this case.

Notice that, as Δ < 0, the square root of the determinant will be an imaginary value. Hence:

Re(x) = -B/2A

Im(x) = ± (√Δ)/2A

An alternative way of dealing with quadratic equations is factoring trinomials . And it really helps if you're able to quickly recognize perfect square trinomials . The next step is to learn how to graph quadratic inequalities .

If, after learning all about solving quadratic equations, you still want more math? Omni has over 240 math calculators to explore. In particular, we recommend you take a look at our cubic equation solver . We also recommend you check out the Computer Technology For Math Excellence's webpage . They have a vast collection of resources to learn all about math, with particular attention to the Common Core curriculum.

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Quadratic equation solver

This step-by-step calculator solves quadratic equations using three different methods: the quadratic formula method , completing the square , and the factoring method . Calculator shows all the work and provides detailed explanation on how to solve an equation.

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How to use this calculator

The most commonly used methods for solving quadratic equations are:

1 . Factoring method

2 . Solving quadratic equations by completing the square

3 . Using quadratic formula

In the following sections, we'll go over these methods.

Method 1A : Factoring method

If a quadratic trinomial can be factored, this is the best solving method.

We often use this method when the leading coefficient is equal to 1 or -1. If this is not the case, then it is better to use some other method.

Example 01: Solve $ x^2 \color{red}{-8}x \color{blue}{+ 15} = 0 $ by factoring.

Here we see that the leading coefficient is 1, so the factoring method is our first choice.

To factor this equation, we must find two numbers ( $ a $ and $ b $ ) with a sum is $ a + b = \color{red}{8} $ and a product of $ a \cdot b = \color{blue}{15} $.

After some trials and errors, we see that $ a = 3 $ and $ b = 5 $.

Now we use formula $ x^2 - 8x + 15 = (x - a)(x - b) $ to get factored form:

Divide the factored form into two linear equations to get solutions.

Method 1B : Factoring - special cases

Example 02: Solve $ x^2 -8x = 0 $ by factoring.

In this case, (when the coefficient c = 0 ) we can factor out $ \color{blue}{x} $ out of $ x^2 - 8x $.

Example 03: Solve $ x^2 - 16 = 0 $ by factoring.

In this case, ( when the middle term is equal 0) we can use the difference of squares formula.

Method 3 : Solve using quadratic formula

This method solves all types of quadratic equations. It works best when solutions contain some radicals or complex numbers.

Example 05: Solve equation $ 2x^2 + 3x - 2 = 0$ by using quadratic formula.

Step 1 : Read the values of $a$, $b$, and $c$ from the quadratic equation. ( $a$ is the number in front of $x^2$ , $b$ is the number in front of $x$ and $c$ is the number at the end)

Step 2 :Plug the values for a, b, and c into the quadratic formula and simplify.

Step 3 : Solve for $x_1$ and $x_2$

Method 2 : Completing the square

This method can be used to solve all types of quadratic equations, although it can be complicated for some types of equations. The method involves seven steps.

Example 04: Solve equation $ 2x^2 + 8x - 10= 0$ by completing the square.

Step 1 : Divide the equation by the number in front of the square term.

Step 2 : move $-5$ to the right:

Step 3 : Take half of the x-term coefficient $ \color{blue}{\dfrac{4}{2}} $, square it $ \color{blue}{\left(\dfrac{4}{2} \right)^2} $ and add this value to both sides.

Step 4 : Simplify left and right side.

Step 5 : Write the perfect square on the left.

Step 6 : Take the square root of both sides.

Step 7 : Solve for $x_1$ and $x_2$ .

1. Quadratic Equation — step-by-step examples, video tutorials with worked examples.

2. Completing the Square — video on Khan Academy

3. Completing the Square — video on Khan Academy

4. Polynomial equation solver

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Quadratic Formula Calculator

The calculator below solves the quadratic equation of ax 2 + bx + c = 0 .

In algebra, a quadratic equation is any polynomial equation of the second degree with the following form:

ax 2 + bx + c = 0

where x is an unknown, a is referred to as the quadratic coefficient, b the linear coefficient, and c the constant. The numerals a , b , and c are coefficients of the equation, and they represent known numbers. For example, a cannot be 0, or the equation would be linear rather than quadratic. A quadratic equation can be solved in multiple ways, including factoring, using the quadratic formula, completing the square, or graphing. Only the use of the quadratic formula, as well as the basics of completing the square, will be discussed here (since the derivation of the formula involves completing the square). Below is the quadratic formula, as well as its derivation.

Derivation of the Quadratic Formula

From this point, it is possible to complete the square using the relationship that:

x 2 + bx + c = (x - h) 2 + k

Continuing the derivation using this relationship:

Recall that the ± exists as a function of computing a square root, making both positive and negative roots solutions of the quadratic equation. The x values found through the quadratic formula are roots of the quadratic equation that represent the x values where any parabola crosses the x-axis. Furthermore, the quadratic formula also provides the axis of symmetry of the parabola. This is demonstrated by the graph provided below. Note that the quadratic formula actually has many real-world applications, such as calculating areas, projectile trajectories, and speed, among others.

Quadratic Formula Calculator

Enter the equation you want to solve using the quadratic formula.

The Quadratic Formula Calculator finds solutions to quadratic equations with real coefficients. For equations with real solutions, you can use the graphing tool to visualize the solutions.

Quadratic Formula : x = − b ± b 2 − 4 a c 2 a

Click the blue arrow to submit. Choose "Solve Using the Quadratic Formula" from the topic selector and click to see the result in our Algebra Calculator !

Solve Using the Quadratic Formula Apply the Quadratic Formula

Popular Problems

Solve Using the Quadratic Formula x 2 + 5 x + 6 = 0 Solve Using the Quadratic Formula x 2 - 9 = 0 Solve Using the Quadratic Formula 5 x 2 - 7 x - 3 = 0 Apply the Quadratic Formula x 2 - 14 x + 49 Apply the Quadratic Formula x 2 - 18 x - 4

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Online Quadratic Formula Calculator

Apply the quadratic formula using wolfram|alpha, a useful tool for finding the solutions to quadratic equations.

2 + b x + c = 0 . In doing so, Wolfram|Alpha finds both the real and complex roots of these equations. It can also utilize other methods helpful to solving quadratic equations, such as completing the square, factoring and graphing.

Learn more about:

• Quadratic formula

Tips for entering queries

Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask about finding roots of quadratic equations.

• quadratic formula 4x^2 + 4 x - 8
• quadratic formula a = 1, b = -1, c = 2
• solve x^2 - x - 4 = 0
• solve x^2 - 3x - 4 = 0
• View more examples

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What are quadratic equations, and what is the quadratic formula?

A quadratic is a polynomial of degree two..

Quadratic equations form parabolas when graphed, and have a wide variety of applications across many disciplines. In physics, for example, they are used to model the trajectory of masses falling with the acceleration due to gravity.

Situations arise frequently in algebra when it is necessary to find the values at which a quadratic is zero. In other words, it is necessary to find the zeros or roots of a quadratic, or the solutions to the quadratic equation. Relating to the example of physics, these zeros, or roots, are the points at which a thrown ball departs from and returns to ground level.

b 2 - 4 a c 2 a   , determines the one or two solutions to any given quadratic. Sometimes, one or both solutions will be complex valued.

Discovered in ancient times, the quadratic formula has accumulated various derivations, proofs and intuitions explaining it over the years since its conception. Some involve geometric approaches. Others involve analysis of extrema. There are also many others. Those listed and more are often topics of study for students learning the process of solving quadratic equations and finding roots of equations in general.

Alternative methods for solving quadratic equations do exist. Completing the square, factoring and graphing are some of many, and they have use cases—but because the quadratic formula is a generally fast and dependable means of solving quadratic equations, it is frequently chosen over the other methods.

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MathPapa - Algebra Calculator 4+

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SOLVE your algebra problems step-by-step with MathPapa! MathPapa can solve your equations (and show the work!) and help you when you're stuck on your math homework. FEATURES: • Solves linear equations and quadratic equations. • Solves linear and quadratic inequalities. • Graphs equations. • Factors quadratic expressions. • Order of operations step-by-step. • Evaluates expressions. • Solves systems of two equations. STEP-BY-STEP SOLUTIONS: MathPapa's goal is to help you learn algebra step-by-step. Get help on your algebra problems with the MathPapa Algebra Calculator! HOW TO USE THE CALCULATOR: Just type your problem into the text box. For example, enter 3x+5=17 into the text box to get a step-by-step explanation of how to solve 3x+5=17. MATH SYMBOLS: Here are some symbols that the calculator understands: + (Addition) - (Subtraction) * (Multiplication) / (Division) ^ (Exponent: "raised to the power") √ (Square Root) |x| (Absolute Value of x) MATHPAPA PREMIUM PRICING AND TERMS: For MathPapa Premium, there is a Monthly subscription for $9.99. This price is for U.S. customers. Pricing in other countries may vary. Subscriptions will be charged via your iTunes account at the end of the free trial. Subscriptions will automatically renew unless canceled in Account Settings at least 24 hours before the end of the current period. Unused portion of Free Trial, if offered, is forfeited after purchase. For more information, read our Terms and Privacy Policy: Terms of Service: https://www.mathpapa.com/terms/ Privacy Policy: https://www.mathpapa.com/privacy/

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The only thing that could save me more is Jesus himself

This app saved me. I was struggling in everything related to algebra. This app not only shows you the answer to fractions, it also shows you the steps to working out. You can't even do what this app does on your phone. Perfect 5 stars. The only way they could make this app better is if they gave the answer in fraction form, word form, and mixed fraction form. I wish I could give this app more than 5 stars because it deserves more. The only in app purchase is a 2$ payment to remove ads, but they really don't bother me that much. I have a feeling I will be using this app all the way through college, and possibly after that. This is the first app in ages I have found that is legit. All of the other apps say they work, but they are actually garbage. Only Jesus could save me more. 5 out of 5, 5 stars, 5 thumbs up, 5 yesseroos, and 5 oh yeahs. Best app on the App Store.

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Thank you so much for your review and feedback. :) So glad you did well in math all throughout college and beyond!

You’ve changed...

How could you do this Math Papa? Since the 7th grade you have helped me survive the endless math homework that was forced upon me, giving me the answer that I needed and the work that went along with it. You were my friend, a great companion for all things math related. But when school rolled around this year, you try to start charging me for the “how to do” part. Why is this? You are a CALCULATOR that was so heavily depended on by many students suffering through their math. And now all of a sudden you get all greedy and take one of the few hopes that we have and pull it out of our grasps. Yes some people might be able to pay for this minor inconvenience and yes you need a way to make a living, but this is not the way. People like me can not and will not pay for certain things like this, having to pay 10 dollars a month for the step by step guide is ridiculous. I’m still keeping the app for a quick conversion here and there, but that’s about it’s good for now. To those who are looking for a free and far superior app, check out Photomath. How could you Math Papa, you’ve changed for the worse...

Dear LBHurd, So glad you enjoyed our step by step homework help and found MathPapa to be a trusty math companion! Yes our prices are slightly higher than your usual “math app.” We've done a lot of work on the app since you first downloaded it. We take extraordinary time and care to craft our explanations and clean design so you can learn and understand your homework with minimal frustration.

Why do we have to pay just for step by step?:(

So from the title you can already tell why I’m writing this. I am a 12 year old girl (I know, I’m probably too little to do this but..) and its my first year in middle school (6th grade) i have LOTS of trouble with math, especially my homework. So I found this app a while back and it said that it gives you answers and step by step. My mom thought it would be a good idea to download this app because I really needed the help, and not to mention that she didn’t understand pre-algebra either. So I downloaded it! It was really good so far until now.. I was on my next question when it said I have to upgrade to premium for steps..and I know my mom wouldn’t pay 10$ just for a step by step. I really liked the step by step because it helped me so much during the 6th grade! I honestly think that this is a great app but you have to pay. And the problem is that I can’t find another app that shows the steps like yours. And the bigger problem is that my homework is due today and I really need help! Do you have any suggestions of what I can do? Thanks!

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• quadratic\:formula\:-x^{2}+4x-3=0
• quadratic\:formula\:-r^{2}-r+2=0
• quadratic\:formula\:4x^{2}=4x-1
• quadratic\:formula\:3x^{2}=2x+5

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• High School Math Solutions – Quadratic Equations Calculator, Part 2 Solving quadratics by factorizing (link to previous post) usually works just fine. But what if the quadratic equation...

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