Quadratic Equation Calculator

The Quadratic Equation Calculator

Quadratic equations are a fundamental part of school and university mathematics curricula. Solving a quadratic equation reveals vital information about a function, including its rates of change, minimums, and maximums. While finding the roots of a quadratic equation requires a standard set of algebraic and arithmetic operations, doing the math manually can be tedious and time-consuming.

Our online quadratic formula calculator is a free, easy-to-use tool that instantly solves quadratic equations. Not only does it provide the final answers, but it also displays the exact steps applied during the calculation. This step-by-step guide helps users fully conceptualize the problem-solving process and understand the numerical results.

Quadratic Equations

A quadratic equation—sometimes referred to as a quadratic function or a second-degree polynomial—is an algebraic equation with the standard form of ax²+bx+c=0, where x is an unknown variable. The terms a and b are the coefficients of x² and x, respectively, while c is a constant. The term "second-degree" refers to the fact that the highest exponent of the variable x is 2. Below are a few examples of quadratic equations:

$$2x²-4x+0.5=0$$

$$-3x²+\frac{1}{3}x+6=0$$

The equation 2x²=0 is also a quadratic equation, where b=0 and c=0. However, 2x+3=0 is not a quadratic equation because it lacks the quadratic term ax². As shown in the examples above, the values of a, b, and c can be positive or negative integers, decimals, or fractions, as long as a≠0.

Solving Quadratic Equations

The number of possible solutions to an algebraic equation equals its highest exponent value. Therefore, a quadratic equation can have a maximum of two solutions (also known as roots). The most reliable way to solve a quadratic function is by using the quadratic formula, as shown in equation (1):

$$x₁=\frac{-b+\sqrt{b²-4ac}}{2a}\ \ \ \ \ \ \ ;\ \ \ x₂=\frac{-b-\sqrt{b²-4ac}}{2a}$$ (1)

The compact form of the quadratic formula is written as:

$$x=\frac{-b±\sqrt{b²-4ac}}{2a}$$

This formula provides a straightforward method: simply plug in the values of a, b, and c to find x₁ and x₂. The number and nature of these solutions depend on the value of the discriminant, which is the expression under the square root, b²-4ac. There are three possible cases:

• If the discriminant is positive (b²-4ac>0), two distinct real solutions exist (x₁≠x₂)
• If the discriminant is zero (b²-4ac=0), one repeating real solution exists (x₁=x₂)
• If the discriminant is negative (b²-4ac<0), two distinct complex solutions exist (x₁≠x₂)

We explore an example of each case in the Examples section below.

Graphically, on an x-y coordinate plane where y is a function of x, the solutions to a quadratic function are the x-intercepts—the exact x-coordinates where the parabola crosses the x-axis.

Using the Quadratic Formula Calculator

Our quadratic solver calculator can easily compute all quadratic equations, regardless of whether the solutions are real or complex. The tool requires three simple inputs: the values of a, b, and c. In some cases, you may need to manipulate your equation into the standard form before using the calculator.

For example, given the equation 2x² = x + 3, you simply move the terms from the right side to the left side. This results in 2x²-x-3=0, where a = 2, b = -1, and c = -3.

Similarly, for an equation like 4(x²-0.2x)=1, you must first expand the parentheses to get 4x²-0.8x=1. Then, move the constant to the left side to achieve the general form 4x²-0.8x-1=0. Here, your inputs would be a = 4, b = -0.8, and c = -1.

Examples

The following three examples illustrate the different possible outcomes when using the quadratic equation calculator.

Example 1: Two Real Solutions

Suppose we need to find the solutions for the quadratic function y₁ given as y₁=x²-8x+12, as shown in Figure 1.

Intuitively, the goal is to find the x-coordinates of the points where the function y₁ crosses the x-axis—if any exist.

Figure 1: Plot of y₁=x²-8x+12

First, equate the function to zero (replacing y₁ with 0) to get x²-8x+12=0. This equation is already in standard form, where a=1, b=-8, and c=12. We can now input these values directly into the quadratic equation formula calculator.

By checking the discriminant, b²-4ac=(-8)²-4(1)(12)=16>0, we confirm that this quadratic function has two real solutions. After clicking the calculate button, the tool instantly provides both the numerical results and the step-by-step breakdown using the standard quadratic formula (1).

It is important to note that after entering the values of a, b, and c, the calculator displays the constructed equation. You should always verify that this matches your intended problem to avoid entry mistakes.

• Equation: x²-8x+12=0

• Solution: x₁=2 and x₂=6

• Steps:

$$x = \frac {-b ± \sqrt{b² - 4ac}}{2a}=\frac{-(-8) ±\sqrt{(-8)^2-4×1×12}}{2×1}=\frac{8 ±\sqrt{16}}{2}=4 ±2=6 \ or \ 2$$

The exact solutions are x₁=2 and x₂=6. We can validate these results graphically by inspecting the parabola's intersection with the x-axis. As shown in Figure 2, the function successfully crosses the x-axis at these precise points.

Figure 2: Plot of y₁=x²-8x+12

Example 2: One Real Solution

Let's consider another function: y₂-3x²+25=-4x²+10x. Before using the calculator, the first step is to isolate y₂ by moving all other terms to the opposite side, resulting in y₂=-4x²+10x+3x²-25. Equating y₂ to zero and simplifying the arithmetic gives us the standard general form: -x²+10x-25=0. Here, a=-1, b=10, and c=-25.

Because the discriminant is exactly zero, b²-4ac=(10)²-4(-1)(-25)=0, we expect a single real solution. Running this through the quadratic formula calculator confirms that x₁=x₂=5.

• Equation: -x²+10x–25=0

• Solution: x = 5

• Steps:

$$x = \frac{-b ± \sqrt{b² - 4ac}}{2a} = \frac{-10±{\sqrt{10^2 – 4 × (-1) × (-25)}}}{2×-1}=\frac{-10± \sqrt{0}}{-2} = 5$$

Figure 3 displays the plot of y₂, clearly showing that the function touches the x-axis at exactly one point.

Figure 3: y₂=-x²+10x-25

Example 3: Two Complex Solutions

Finally, let's examine the function y₃=x²-4x+8 to see how a quadratic equation can yield two complex solutions. As illustrated in Figure 4, the parabola for y₃ never intersects the x-axis.

Figure 4: y₃=x²-4x+8

Calculating the discriminant gives us b²-4ac=(-4)²-4(1)(8)=-16<0. A negative discriminant proves the existence of two complex solutions. But what exactly is a complex number?

A complex number is a combination of real and imaginary numbers, typically expressed in the form a+ib.

In this format, 'i' stands for the imaginary unit, which represents the square root of -1.

The term a denotes the real part of the complex number (Re). On the other hand, ib represents the imaginary part (Im), where i=√-1.

Whenever the discriminant b²-4ac is less than zero, the quadratic formula requires taking the square root of a negative number, which is only possible using complex numbers.

Returning to our equation x²-4x+8=0, the calculator efficiently solves the problem and provides the roots x₁=2+2i and x₂=2-2i.

• Equation: x²–4x+8=0

• There are two possible solutions: x=2±2i

• Steps:

$$x = \frac{-b ± \sqrt{b² – 4ac}}{2a} = \frac{-(-4) ± \sqrt{(-4)^2 – 4 × 1 × 8}}{2 × 1} = \frac{4 ± \sqrt{-16}}{2} = 2 ± 2i$$

Scope of Use and Tips

Our quadratic formula calculator is optimized for school and university students, professionals, or anyone seeking a rapid, reliable solution to quadratic functions. These equations frequently appear across various fields, including engineering, economics, physics, and agriculture.

While our online solver is highly intuitive, users should be comfortable performing basic arithmetic to rearrange their equations into the standard ax²+bx+c=0 format. Additionally, a basic understanding of complex numbers is helpful—though not strictly required—since quadratic roots occasionally emerge as complex pairs.

For deeper insights, users may also want to combine this calculator with graphic plotting tools to visually verify the parabola and precisely locate its x-intercepts.