Which values should I enter?

The required values depend on the selected form. Enter $a$, $b$, and $c$ for standard form; $a$, $h$, and $k$ for vertex form; and $a$, $p$, and $q$ for factored form.

How are the roots found in factored form?

In an equation of the form $a(x - p)(x - q) = 0$, the roots are directly $x = p$ and $x = q$.

How do you solve a quadratic equation?

If a quadratic equation is given in standard form, begin by calculating the discriminant. Then apply the quadratic formula according to the discriminant's value to find the equation's real roots.

What is the quadratic equation formula?

The most commonly used method for solving quadratic equations is the quadratic formula: $x = \frac{-b \pm \sqrt{\Delta}}{2a}$ The discriminant in the formula is calculated using $\Delta = b^2 - 4ac$.

What does the discriminant mean?

The discriminant indicates how many real roots the equation has. If it is positive, there are two real roots; if it is zero, there is one; if it is negative, there are no real roots.

What happens if $a = 0$?

When $a = 0$, the equation is no longer quadratic. The $x^2$ term disappears, and the equation becomes linear.

What should I do if a term is missing?

Use $0$ as the coefficient of the missing term. For example, in $x^2 - 16 = 0$, $b = 0$.

What is the difference between vertex form and standard form?

Standard form clearly shows the equation's coefficients. Vertex form directly shows the parabola's vertex, namely $(h, k)$.

What does “no real roots” mean?

It means the equation has no solution in the set of real numbers. On the graph, the parabola does not intersect the x-axis.

Can calculations be performed with decimal coefficients?

Yes. Coefficients may be integers, fractions, or decimals. However, calculations with decimal values may produce small rounding differences.

What is the most common mistake?

The most common mistake is entering the coefficients before converting the equation to standard form. Make sure negative signs are entered correctly.

Quadratic Equation Solver

Calculate the discriminant and real roots from standard, vertex, or factored form.

Quadratic equations may look a little crowded at first glance. It can be difficult to know where to begin, especially when $a$, $b$, $c$, the discriminant, and the root formula all appear at the same time. This calculator determines the discriminant, the number of real roots, and the roots based on the form in which you enter the equation.

You can write the equation in standard form. If you have the vertex form, you can use that as well. If the equation is already factored, you can also see the roots directly.

Entering the Equation

The calculator works with three different equation forms.

In standard form, the equation is written as follows:

Formül

$$ax^2 + bx + c = 0$$

a = ikinci dereceli terimin katsayısıb = birinci dereceli terimin katsayısıc = sabit terimx = bilinmeyen

When you select this form, you must enter the values of $a$, $b$, and $c$.

The vertex form is written as follows:

Formül

$$a(x - h)^2 + k = 0$$

a = parabolün açılma yönünü ve genişliğini belirleyen katsayıh = tepe noktasının x koordinatık = tepe noktasının y koordinatıx = bilinmeyen

This form uses the values of $a$, $h$, and $k$.

The factored form is written as follows:

Formül

$$a(x - p)(x - q) = 0$$

a = çarpanların önündeki katsayıp = birinci kökq = ikinci kökx = bilinmeyen

In this form, the roots can often be read directly from the equation. However, the coefficient $a$ still affects the discriminant calculation.

What Should We Enter into the Calculator?

If the equation is $x^2 - 5x + 6 = 0$, select the standard form. The values in this example are:

$a = 1$, $b = -5$, $c = 6$

If the equation is $(x - 2)^2 - 1 = 0$, the vertex form is more suitable:

$a = 1$, $h = 2$, $k = -1$

If the equation is $(x - 2)(x - 3) = 0$, you can use the factored form:

$a = 1$, $p = 2$, $q = 3$

There is a small sign detail to note here. In factored form, $x + 1$ means $x - (-1)$. Therefore, if the equation contains the factor $x + 1$, the corresponding root is $-1$.

When Is an Equation Quadratic?

For an equation to be quadratic, the highest power of the unknown must be 2. The general form is again:

Formül

$$ax^2 + bx + c = 0$$

a = ikinci dereceli terimin katsayısıb = birinci dereceli terimin katsayısıc = sabit terimx = bilinmeyen

The coefficient $a$ cannot be zero. If $a = 0$, the $x^2$ term disappears. The equation is then no longer quadratic; it becomes linear.

For example, $3x^2 - 2x + 1 = 0$ is quadratic. However, if it is written as $0x^2 - 2x + 1 = 0$, only $-2x + 1 = 0$ remains.

What Does the Discriminant Show?

For an equation in standard form, the discriminant is calculated first to examine the roots:

Formül

$$\Delta = b^2 - 4ac$$

\Delta = diskriminant değeri

The sign of the discriminant determines the number of real roots:

Formül

$$N = \begin{cases} 2, & \Delta > 0 \\ 1, & \Delta = 0 \\ 0, & \Delta < 0 \end{cases}$$

N = gerçek kök sayısı\Delta = diskriminant değeri

Therefore, if $\Delta > 0$, there are two distinct real roots. If $\Delta = 0$, the roots coincide. If $\Delta < 0$, the equation has no solution within the real numbers.

The roots are calculated using the following formula:

Formül

$$x = \frac{-b \pm \sqrt{\Delta}}{2a}$$

x = denklemin kökü

Roots in Vertex Form

If the equation is in vertex form, the roots can be found directly using the following formula:

Formül

$$x = h \pm \sqrt{\frac{-k}{a}}$$

x = denklemin kökü

The expression inside the square root is important in this form. If $\frac{-k}{a}$ is negative, there are no real roots.

The discriminant can also be calculated in vertex form as follows:

Formül

$$\Delta = -4ak$$

\Delta = diskriminant değeri

This form is particularly useful when the vertex of the parabola is known. There is no need to expand the equation into standard form; the result can be found directly from the values of $a$, $h$, and $k$.

Roots in Factored Form

If the equation is given in factored form, the process is shorter:

Formül

$$a(x - p)(x - q) = 0$$

a = çarpanların önündeki katsayıx = bilinmeyenp = birinci kökq = ikinci kök

For a product to equal zero, at least one of its factors must be zero. Therefore, the roots are:

Formül

$$x_1 = p,\quad x_2 = q$$

x_1 = birinci kökx_2 = ikinci kök

To determine the discriminant, use the following formula:

Formül

$$\Delta = a^2(p - q)^2$$

\Delta = diskriminant değeri

The leading coefficient $a$ does not change the positions of the roots. However, it affects the vertical shape of the parabola.

Let Us Work Through a Few Examples

Consider the following equation first:

$$x^2 - 5x + 6 = 0$$

Here, $a = 1$, $b = -5$, and $c = 6$.

Let us calculate the discriminant:

$$\Delta = (-5)^2 - 4 \times 1 \times 6 = 25 - 24 = 1$$

Because $\Delta = 1$ is positive, we expect two distinct real roots.

The root formula is:

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

The first root is:

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

The second root is:

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

The roots of this equation are $x = 3$ and $x = 2$.

Now consider an example in which the roots coincide:

$$x^2 - 4x + 4 = 0$$

The coefficients are $a = 1$, $b = -4$, and $c = 4$.

$$\Delta = (-4)^2 - 4 \times 1 \times 4 = 16 - 16 = 0$$

Because the discriminant is zero, both roots have the same value:

$$x = \frac{-(-4)}{2 \times 1} = \frac{4}{2} = 2$$

In this case, the repeated root is $x = 2$. On the graph, the parabola is tangent to the $x$-axis at this point.

Now consider a case with no real roots:

$$x^2 + 2x + 5 = 0$$

Here, $a = 1$, $b = 2$, and $c = 5$.

$$\Delta = 2^2 - 4 \times 1 \times 5 = 4 - 20 = -16$$

Because $\Delta < 0$, there are no real roots. The parabola does not intersect the $x$-axis.

Example Using Vertex Form

Consider the equation:

$$(x - 2)^2 - 1 = 0$$

Here, $a = 1$, $h = 2$, and $k = -1$.

Apply the root formula:

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

$$x = 2 \pm \sqrt{1}$$

Therefore:

$$x_1 = 2 + 1 = 3$$

$$x_2 = 2 - 1 = 1$$

We can also verify the same result using the discriminant:

$$\Delta = -4 \times 1 \times (-1) = 4$$

Because the discriminant is positive, there are two distinct real roots.

Now consider a vertex-form equation with no real roots:

$$(x - 2)^2 + 1 = 0$$

This time, $a = 1$, $h = 2$, and $k = 1$.

The expression inside the square root is:

$$\frac{-k}{a} = \frac{-1}{1} = -1$$

A negative number has no square root within the real numbers. Therefore, the equation has no real roots.

The discriminant gives the same result:

$$\Delta = -4 \times 1 \times 1 = -4$$

Because $\Delta < 0$, there are no real roots.

Example Using a Factored Equation

In the following equation, the roots are almost immediately visible:

$$(x - 2)(x - 3) = 0$$

One of the factors must equal zero:

$$x - 2 = 0 \Rightarrow x = 2$$

$$x - 3 = 0 \Rightarrow x = 3$$

Here, $a = 1$, $p = 2$, and $q = 3$.

The discriminant is:

$$\Delta = 1^2(2 - 3)^2 = 1$$

The roots are $x = 2$ and $x = 3$.

An example with a leading coefficient is:

$$5(x + 1)(x - 4) = 0$$

We can interpret $x + 1$ as $x - (-1)$:

$$5(x - (-1))(x - 4) = 0$$

In this case, $a = 5$, $p = -1$, and $q = 4$.

The roots are:

$$x_1 = -1$$

$$x_2 = 4$$

The discriminant is:

$$\Delta = 5^2(-1 - 4)^2 = 25 \times 25 = 625$$

The coefficient $5$ does not change the roots. The roots still come from the factors. The coefficient mainly determines how narrow or wide the graph appears vertically.

Interpreting the Results

If $\Delta > 0$, the equation has two distinct real roots. On the graph, the parabola intersects the $x$-axis at two separate points.

If $\Delta = 0$, there is one real root value. In fact, the two roots coincide at the same point. The parabola is tangent to the $x$-axis.

If $\Delta < 0$, there are no real roots. The equation can be solved using complex numbers, but if the calculator displays only real roots, the results section will state that no real roots exist.

To verify a root, substitute its value back into the original equation.

For example, let us test $x = 2$ in the equation $x^2 - 5x + 6 = 0$:

$$2^2 - 5 \times 2 + 6 = 4 - 10 + 6 = 0$$

Because the equality is satisfied, $x = 2$ is indeed one of the roots of this equation.

Points to Consider When Using the Calculator

Do not enter zero for $a$. In that case, the equation is not quadratic.
When working in standard form, move all terms to the same side of the equation.
Enter negative coefficients together with the minus sign.
Treat the coefficient of any missing term as $0$.
In vertex form, pay attention to the sign in the expression $x - h$.
In factored form, $x + 1$ means $x - (-1)$.
Small rounding differences may occur with decimal coefficients.
A “no real roots” result is often not an error; the discriminant may be negative.

Where Is It Useful?

These equations do not appear only in algebra exercises. Quadratic equations are also used in parabola graphs, motion problems, area problems, and maximum-minimum value calculations.

You can use this calculator especially in the following situations:

Checking calculations performed with the quadratic formula
Finding where a parabola intersects the $x$-axis
Verifying a factorization result
Solving equations that arise in trajectory or motion problems in physics
Quickly checking answers to exam questions
Observing how different coefficients change the roots

When solving exercises, using the calculator to verify the result rather than simply obtain it is more instructive. Solving the problem yourself first and then entering the values into the calculator makes it easier to identify the step where an error occurred.

Quadratic Equation Solver

Entering the Equation

What Should We Enter into the Calculator?

When Is an Equation Quadratic?

What Does the Discriminant Show?

Roots in Vertex Form

Roots in Factored Form

Let Us Work Through a Few Examples

Example Using Vertex Form

Example Using a Factored Equation

Interpreting the Results

Points to Consider When Using the Calculator

Where Is It Useful?

How we tested it

Frequently Asked Questions

References and Sources