Introducing Expanded Step-by-step Math Solutions

Step-by-step solutions, one of the most popular features for mathematics in Wolfram|Alpha, has just received a dramatic expansion in its functionality! With our new interface, you now have the ability to walk through all of our Step-by-step solutions at your own pace, revealing only one step at a time. Some of our programs will offer to guide you with hints when walking through solutions. And for common math problems, we can even show multiple ways to find the solutions. We are also very excited to introduce three new math content areas that now have Step-by-step solutions: solving equations, rational arithmetic, and verifying trigonometric identities. When you’re signed into Wolfram|Alpha, you can use this new feature three times a day. Or, when you upgrade to Wolfram|Alpha Pro, you can use it as many times as you like!

Let’s look at a new Step-by-step solution for an integral (one of the more popular math queries we receive). We’ll type “integrate cos^2(x)” into Wolfram|Alpha and then click the Step-by-step solution button in the top right of the results page.

To walk through the problem one step at a time, you can click the Next step button, as we have done above. Or if you’d rather see everything at once, click the Show all steps button:

Now let’s look at the input (8 * 11) / 3 + 4, which features a Step-by-step solution from one of our brand-new programs. In this walkthrough, you will have the option to use hints to help guide you through the problem:

As you walk through the problem, hints will give you an idea of what comes next. If you’d rather not use the hints, you can click the Hide hints button in the top right. And of course, if you’d like to see all of the steps at once, we can click “Show all steps,” as we did in our first example.

Wolfram|Alpha’s capability to show steps to solve an equation has grown tremendously over the summer! To see this, let’s start by finding the roots of a polynomial:

The top-right corner of the Step-by-step solutions window has a drop-down menu to let us choose how to solve the problem: use the factor method, complete the square, or use the quadratic formula. Let’s try all three and compare:

Again, we see that we have the option to walk through the steps one at a time (using hints if we’d like) or to show all steps at once.

In addition to offering hints and multiple methods to solve a problem, we can now solve equations over the real numbers or over the complex numbers! Let’s see this in action by asking Wolfram|Alpha to find the roots of (e^x + 2)(x – 1). When solving over the real numbers, Wolfram|Alpha will show us that (e^x + 2)(x – 1) has only one root; over the complex numbers, Wolfram|Alpha will find the complex roots of this expression.

To see even more of our brand-new functionality, let’s ask Wolfram|Alpha to verify a trigonometric identity. To do this, we simply type the identity we wish to prove into Wolfram|Alpha, and it will walk us through our proof one step at a time. For example, let’s try the identity (sin(x) – tan(x))(cos(x) – cot(x)) = (sin(x) – 1)(cos(x) – 1):

Here are some more examples for you to explore the scope of Step-by-step solutions.

Arithmetic:

• 60431 / 89
• 6876 + 9782 + 816
• 9 (3+1) + 17 / (6-12)

Equation solving:

• 4x – 6 = 2x+8 solution
• solve (9^(x + 1)) – (28 (3^(x))) + 3 = 0 over the real numbers
• solve: x^4 + 2x^3 + 5x^2 + 10x + 25 = 0

Polynomial expansion:

• expanded form of (x + 3)^2
• expand (x + 1)(x – 1)(x + 2)
• multiply out (x^2 + x)^4

Partial fraction decomposition:

• partial fraction decomposition 1/(x^2 + 4x + 3)
• 1 / (x^3 + 4x^2 + 5x + 2)
• 1 / (x^4 + 8x^3 + 22x^2 + 24x + 9) partial fraction expansion

Matrix row reduction:

• {{1,1,5},{1,-1,1}} row reduce
• reduced row echelon form: {{1, -3, 3, -4}, {2, 3, -1, 15}, {4, -3, -1, 19}}

Proving trigonometric identities:

• (1 + tan(x))/(1 – tan(x)) = (cos(x) + sin(x))/(cos(x) – sin(x))
• sin(x)^4 – cos(x)^4 = 1 – 2 cos(x)^2
• cot(t/2)^2 = (1 + cos(t)) / (1 – cos(t))

Limits:

• limit of (x – 3) / (x^2 – 2x – 3) as x approaches 3
• (e^x – 1 – x) / x^2 as x goes to 0
• take the limit as x goes to infinity: (1 + 1/x)^x

Derivatives:

• derivative of x^4 + 9x^3 + 7x – 2
• sin(square root of x) derivative
• slope of log(x) / (x^2 + 1)

Integrals:

• integrate sin(x)cos(x)^2
• integral of sqrt(a^2 – x^2)
• show the integration: arcsec(sqrt(t))

Ordinary differential equations:

• y‘(t) – 2y(t) = 3 e^(2t)
• y”(t) + y(t) = sin(t)
• t^2 y‘(t) + 2t y(t) = t^4 y(t)^2 + 4

This gives you a brief overview of what you can do with our new Step-by-step solutions. When you’re signed into Wolfram|Alpha, you can use this new feature three times a day. Wolfram|Alpha Pro users receive unlimited access to Step-by-step solutions.

With Wolfram|Alpha’s Step-by-step Solutions feature, you can be guided—at your own pace—through a broad range of math problems, from arithmetic and equation solving all the way through integrals and ordinary differential equations. We look forward to expanding our Step-by-step solutions to more areas—please let us know if there are new solutions that you’d like to see!