Quadratic Formula Calculator: Find the Roots of any Quadratic Equation

How to Use Our Quadratic Formula Calculator

To solve your equation, you first need to make sure it’s in the standard form: ax2+bx+c=0. Once you have it in this format, simply identify the coefficients a, b, and c.

• Enter the value for ‘a’: This is the number in front of the x2 term. Note that a cannot be 0.

• Enter the value for ‘b’: This is the number in front of the $x$ term. If there is no x term, b is 0.

• Enter the value for ‘c’: This is the constant term (the number without an x). If there is no constant, c is 0.

Once you’ve entered the three coefficients, click “Calculate” to see the solutions.

Understanding Your Results

The calculator provides the value(s) of x that make the equation true. These solutions are called the roots or zeros of the equation. Geometrically, they are the points where the graph of the quadratic equation (a parabola) intersects the x-axis.

The Quadratic Formula

Our calculator uses the famous quadratic formula to find the solutions:

x=frac−bpmsqrtb2−4ac2a

Let’s break down the components of this formula:

• The Discriminant (b2−4ac): This part of the formula under the square root is the most important piece of information. It “discriminates” or determines the nature of the roots without you having to solve the entire equation.

  • If b2−4ac0 (positive): There are two distinct real roots. This means the parabola crosses the x-axis at two different points.

  • If b2−4ac=0: There is exactly one real root. The vertex of the parabola touches the x-axis at a single point.

  • If $b^2 – 4ac \< 0$ (negative): There are no real roots. The solutions are two complex conjugate roots. This means the parabola never touches or crosses the x-axis.

• The ± Symbol: This indicates that there are two potential solutions. One is found by adding the square root of the discriminant, and the other is found by subtracting it.

  • x_1=frac−b+sqrtb2−4ac2a

  • x_2=frac−b−sqrtb2−4ac2a

Frequently Asked Questions

What is a quadratic equation?

A quadratic equation is a polynomial equation of the second degree, meaning it contains a term where the variable is raised to the power of 2. The standard form is ax2+bx+c=0, where a, b, and c are coefficients and a is not equal to 0. The graph of a quadratic equation is a U-shaped curve called a parabola.

What are the different ways to solve a quadratic equation?

There are three primary methods:

1. Factoring: This involves rewriting the equation as a product of two linear expressions. It’s fast but only works for some equations.

2. Completing the Square: A method to convert the equation into a perfect square trinomial. It works for all equations but can be complex.

3. The Quadratic Formula: This is the universal method. It works for every quadratic equation, making it the most reliable tool.

When should I use the quadratic formula?

You should use the quadratic formula when you can’t easily solve the equation by factoring, or when you want a direct and guaranteed method to find the solution. It is especially necessary when the roots are irrational or complex.

What does the graph of a quadratic equation look like?

The graph is always a parabola.

• If the coefficient a is positive, the parabola opens upwards (like a smile).

• If the coefficient a is negative, the parabola opens downwards (like a frown). The roots of the equation are the points where this parabola crosses the horizontal x-axis.

What are complex or imaginary roots?

When the discriminant (b2−4ac) is negative, you can’t take its square root using real numbers. This is where imaginary numbers come in, defined by i, where $i=sqrt-1$. The solutions will be in the form of p ± qi, where p and q are real numbers. Geometrically, this means the parabola does not intersect the x-axis at all.

Concrete Example: Solving a Real-World Problem

Let’s say you throw a ball upwards. The height h of the ball after t seconds can be modeled by the quadratic equation: h(t)=−16t2+48t+4. You want to find out how long it takes for the ball to hit the ground.

1. Set up the Equation: Hitting the ground means the height h is 0. So, we need to solve: −16t2+48t+4=0.

2. Identify Coefficients:

   • a = -16

   • b = 48

   • c = 4

3. Use the Calculator: Enter these values for a, b, and c.

4. Result: The calculator will provide two solutions for t: one positive and one negative.

   • $t\_1approx3.08$ seconds

   • $t\_2approx-0.08$ seconds

5. Interpret the Answer: Since time cannot be negative in this context, we discard the negative root. It will take approximately 3.08 seconds for the ball to hit the ground.

What if the ‘b’ or ‘c’ term is missing?

The calculator can still solve it.

• If b=0 (e.g., 2x2−32=0): Enter a=2, b=0, c=-32.

• If c=0 (e.g., 3x2+6x=0): Enter a=3, b=6, c=0.

Why can’t ‘a’ be zero?

If a were 0, the ax2 term would disappear, and the equation would become $bx+c=0$. This is a linear equation, not a quadratic equation, and it only has one solution, which can be found with simple algebra ($x=-c/b$).

What is the “vertex” of a parabola?

The vertex is the highest or lowest point of the parabola. Its x-coordinate can be found with the simple formula $x=-b/2a$. Once you have the x-coordinate, you can plug it back into the original equation to find the y-coordinate of the vertex.

Can I just guess and check to find the roots?

For very simple equations with integer roots (like x2−4=0), guessing might work. However, for most quadratic equations, especially those with decimal or irrational roots, guessing is highly impractical and inefficient. The quadratic formula provides a precise and systematic way to find the exact answer every time.

Now that you’ve solved your quadratic equation, you might want to visualize it. Our upcoming Graphing Calculator can help you plot the parabola. For other algebraic needs, check out our Factoring Calculator or our general-purpose Scientific Calculator.