Quadratic Formula Calculator

Quadratic Formula Calculator With Full Working Shown

This quadratic formula calculator is free and will show you every step of the process. You don’t even have to sign up or pay. As you enter a, b, and c, you’ll be able to see the discriminant, both of the roots, and the whole procedure behind each one. You can easily copy this material for your own activity or lesson plan. 

Below we don’t simply provide you a formula and leave it at that. First we’ll show you how to understand the discriminant, which tells you what kind of answer you may predict even before you start to solve. Then we go over three alternative approaches to solve the problem, and we show you each method on an equation where it truly works best, not the same example that was used in all three just to check a box.

Start Here: What the Discriminant Tells You

Before you try to solve a quadratic equation, you need to do one math problem on a quadratic formula calculator. Which method you pick might not work because it tells you what kind of answer you can expect. It’s called the forecast.

D = b² − 4ac

You don’t have to do anything else because this number tells you right away how many real solutions your equation has:

Let’s use a real-world example: x² – 5x + 6 = 0. In this case, a = 1, b = -5, and c = 6.

D = -5² – 4(1)(6) = 25 – 24 = 1

Before we do anything else, we know that this equation has two separate real roots because D is positive. A lot of the time, checking the discriminant first keeps you from getting lost. That way, you’ll never be surprised by a hard answer or wonder why factoring didn’t give you a clean second root.

Before you write anything else, do this one little check. It will save you a lot of time on tests and tasks. If students don’t do it, they often waste time factoring an quadratic equation calculator that will never factor cleanly and then move on to the quadratic formula calculator. It only takes ten seconds to figure out D first, which tells you exactly what kind of answer you’re going for and which method you should try.

Route 1: Factoring (Fastest When It Works)

Some things can be done faster with factoring, but not all of them. We already knew the answer: x² – 5x + 6 = 0.

Find two numbers that add up to -5 and multiply them to get c, which is 6. They are not the same because (−2) × (−3) = 6 and (−2) + (−3) = -5.

(x – 2)(x – 3) = 0

Setting each value to 0 gives you both roots:

x – 2 = 0 → x = 2; x – 3 = 0 → x = 3

You have ten seconds to find the right pair of numbers. If you can’t, stop and try again another way. As I worked on these problems, I learned this. For the roots to work quickly, they need to be whole numbers or clean parts. It is a waste of time to try to solve a problem that you can’t. Move on to a method that always works instead.

Route 2: Completing the Square (Best for Vertex Form)

After factoring the square, you have to do a little more work to finish it. How to calculate quadratic formula this gets you the free vertex of the curve. x² + 6x + 5 = 0 is a different one.

First, move the constant to the other side:

-5x² plus 6x equals

Next, take the square root of half of b, which is 6, and add it to both sides. Three times six is three, and three times three is nine.

-5 + 9 x² + 6x + 9 = 4

Now the left side is a square:

(x + 3)² = 4

Take the square root of each side. Remember to add both the positive and negative roots:

x + 3 = ±2

Getting through both:

x + 3 = 2 → x = -1 x + 3 = -5 → x = -8

You get more than just the two roots with this method. The equation now looks like this: (x + 3)² = 4. This shows that the point of the parabola is at x = -3. That’s not something that factoring ever tells you directly.

Route 3: The Quadratic Formula (Always Works)

But you can only perform so much algebra or complete the square. The quadratic technique is incorrect. It can be used for any quadratic formula solver, even those whose roots are difficult to understand or do not exist as real numbers.

x = (-2a)/(-4ac ± b^2)

2×2 + 3x – 4 = 0 doesn’t function well with factors. This time a = 2, b = 3, c = -4.

What makes you distinctive? First, identify:

D = b² – 4ac = 3² – 4*2*(-4) = 41

This equation does not hold true because we already know that √41 is not a whole number. That’s why we made the proper decision. To summarize, use the formula:

x = (-3 ± √41) / (2 × 2) x = (-3 ± √41) / 4 We observe that 6.403 is near to √41, therefore x = (−3 + 6.403) ÷ 4 = 3.403 ÷ 4 ≈ 0.851, and x = (−3 − 6.403) ÷ 4 = −9.403 ÷ 4 ≈ -2.35

You can try this procedure with your own equation putting in your own a, b, and c numbers, the same as given below. Whatever is done for the substitution is evident.

When the Discriminant Is Negative: Complex Roots

That said, it’s not always inaccurate when the computer comes up empty. There is no definitive answer to the equation for this question. Let’s find the solutions of the equation x² + 2x + 5 = 0, where a = 1, b = 2, and c = 5.

D = b² – 4ac = 2² – 4(1)(5) = 4 – 20 = -16

Since D is negative, its square root gives you the imaginary unit i, with i = -1.

Let us accomplish this step by step: x = (-2 +/- sqrt(-16) / 2  x = (-2 +/- 4i) / 2 x = -1 +/- 2i So there is no real number $x$ for which this equation is true. What does this signify in mathematics? It signifies that the curve of this equation never crosses the x-axis from above or from below. It is also helpful to know that the complex roots of a quadratic formula calculator always come in pairs, which make the roots equal. This suggests that if $-1+2i$ is a solution, then $-1-2i$ is one as well. You cannot get a single complicated root by itself.

Which Route Should You Use?

Here’s a direct comparison of all three methods, based on what each one is actually good for:

Look at the discriminant first, then try factoring for ten seconds. This rule can help you solve almost any equation. That should work. If not, go straight to the quadratic formula calculator. Just in case the faster ones don’t work out right away, this one always does.

Seeing It on a Graph

Clearly we see that there is a parabola where ax² + bx + c = 0. When you look for roots, that is all you will discover. The x-intercepts are the points where the parabola intersects the x-axis.

The three graphs of parabolas were drawn side by side to illustrate that D > 0 contacted the x-axis once, D = 0 crossed it twice, and D < 0 did not touch it.

What is the top or bottom of the parabola? It is at x = -2a. Also, it is the number that you get after you complete the square with the help of a quadratic formula calculator. If the vertex of the parabola is entirely above or below the x-axis, the curve never reaches the axis. So there is no genuine x-value where it crosses. This is consistent with the fact that if D < 0, then there are no solutions. That’s the graphic for the difficult roots scenario in the last part.

FAQs

Q1. What is the quadratic formula?

If you need to solve an equation like ax² + bx + c = 0, you can use x = (−b ± √(b² − 4ac)) ÷ 2a. The answer is −2.351 or x = 0.851 when you solve 2x² + 3x − 4 = 0.

Q2. What does the discriminant tell you?

Do not try to solve an equation until you know how many real solutions it has. This can be done with D = b² – 4ac. If D is greater than 0, there are two different real roots. If D = 0, there is one real root that shows up more than once. A real answer can’t be found if D is less than 0. This means the roots are not simple.

Q3. When should I factor instead of using the formula?

First, try to factor it and see how long it takes you to find the right pair of numbers. Is there something that doesn’t make sense right away? Use the quadratic formula calculator instead of factoring. Plus, it works all the time, even when factoring doesn’t.

Q4. What does it mean when there are no real solutions?

A negative coefficient and large roots, such as -2i − 1, are what it means. When you look at a graph, this means that the parabola never goes below or above the x-axis.

Q5. Can a quadratic have only one solution?

Yes. If the discriminant is exactly 0, the equation has one real root that shows up more than once. There is only one touch between the parabola’s point and the x-axis, so it does not cross it twice.

Q6. Is this calculator’s step-by-step working really free?

Yes. For any of the steps on this page or in the tool above, you don’t have to sign up, become a member, or pay. This is not like some other solvers that need you to sign up or pay before they can fully work.