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Zero Product Property
Why a quadratic equation or any other equation must be equal to zero in order to find out its roots or solutions?� When a system of linear equations has no constant term, they are called homogeneous. This essentially means that all terms have the same degree (you can also have homogeneous equations of higher degree, e.g. quadratic equations in which all terms have degree 2). In general, if you multiply all variables of homogeneous equations by a constant the right hand sides are still zero. One solution is that all variables are zero, but if you find one non-zero solution, all multiples of that solution are solutions. Does that apply here? No because the first equation says y = 2x/3 and the second says y. Education. Math. Algebra. Solving Equations with Complex Solutions. Solving Equations with Complex Solutions. In this article. By Mary Jane Sterling. You often come across equations that have no real solutions � or equations that have the potential for many more real solutions than they actually have. For instance, the equation x 2 + 1 = 0 has no real solutions. If you write it as x 2 = �1 and try to take the square root of each side, you run into trouble. Not until you have the imaginary numbers can you write that the solution of this equation is x = +/� i. The equation has two complex solut. To find the zeros of function f, solve the equation f(x) = -2x + 4 = 0 Hence the zero of f is give by x = 2. Example 2. Find the zeros of the quadratic function f is given by.� Solution to Example 5. Solve f(x) = 0 ex2 - 2 - 3 = 0 Rewrite the above equation as follows ex2 - 2 = 3 Rewrite the above equation changing it from exponential to logarithmic form x2 - 2 = ln (3) Solve the above equation to find two zeros of f x1 = square root [ln (3) + 2] and x2 = - square root [ln (3) + 2].

There are three solution types that can cause confusion. We'll look at one example of each, and I'll explain the differences. Then we'll work on a mixture of equation type, so that you can become more comfortable in telling the solution types apart. To solve this equation, I first need to simplify the left-hand side by taking the "minus" through the parenthetical, and All Equations For Maths Gcse Question combining "like" terms:.

Yes, indeed, it is, because zero is a valid number. It's not that the solution is "nothing"; it's that the solution is "something", and that this "something" is zero. So my answer is:. Students can generally become comfortable with zero being the solution to an equation, but the difference between a solution of "zero" that solution being a numerical value and "nothing" being possibly a physical measure of something like "no apples" or "no money" can cause confusion.

Please make sure that you understand that "zero" itself is not "nothing". Zero is a numerical value which in "real life" or in the context of a word problem might imply that there is "nothing" of something or other, but zero itself is a real thing; it exists; it is "something".

Since when is four ever equal to five? Is there any possible x -value that will "fix" this equation, to make it say something that makes any sense? Will any value of x ever make this equation work?

No; it's simply not possible. I did all of my steps correctly, but those steps led to an equation a contained no variable and b made no sense.

Since there is no x -value that will make this equation work, then there is no solution to this equation. And that's my answer for this exercise:. Here's the logic for the above example: When you try to solve an equation, you are starting from the unstated assumption that there actually is a solution. Advisory: This answer is entirely unlike the answer to the first exercise at the top of this page, where there was All Equations For Maths Gcse Levels a value of x that would work that solution value being zero.

Don't confuse these two very different situations: "the solution exists and has the value of zero" is not in any manner the same as "no solution value exists at all". And don't confuse the "no solution" type of equation above with the following type of equation:.

This result is the opposite of that. For this equation, is there any possible value of All Equations For Maths Gcse 3d Pdf x that could make the above statement false? No; 5 is always All Equations For Maths Gcse Online Booking going to be equal 5. In fact, since there is no " x " in the last line of computations above, the value of x is clearly irrelevant to the equation; x can be anything I want, and the equation will still be true.

So the solution is:. You should expect to see some variation in lingo from one textbook or instructor to the next, so don't be surprised at differences in formatting. Note that, if I had solved the equation by subtracting a 5 from either side of the original equation, I would have ended up with:.

In other words, I would have ended up with another trivially-true statement. I also could have subtracted 4 x from either side, or I could have divided both sides of the above equation by 4 , or I could have divided through by 4 and then subtracted x from either side, or I could have subtracted both 4 x and 5 from both sides of the original equation. But no matter the specific steps taken, the result a trivially-true equation will always be the same, and solution will still be the same: "all x ".

Since as I've listed above there are many ways of arriving at the same conclusion for this type of equation, you should not be surprised if, for "all real numbers" or "no solution" equations, you didn't use the exact same steps as some of your classmates.

Unfortunately, while you'll almost certainly see at least one of these "no solution" or "all reals" questions on the next test and probably also on the final , there usually aren't many in the homework set, and your instructor probably provided only one example of each type. That doesn't give you much practice at interpreting these types of solutions, so let's so some more examples. First, I'll multiply the 3 through the parenthetical on the left-hand side.

Then I'll solve. My math was correct, but the result is nonsense. Twelve is never going to equal eleven. Zero is always going to be equal to zero, and there's not even any variable in the last line of my work, so the variable is clearly irrelevant.

This equation is true, regardless of the value of x. Page 1 Page 2 Page 3 Page 4 Page 5. All right reserved. Web Design by. Skip to main content.

Purplemath There are three solution types that can cause confusion. Content Continues Below. First, combine like terms; then solve:. First, I'll combine like terms; then I'll solve:. I'll multiply through and simplify on the left-hand side. I'll need to multiply through and simplify on each side of this equation.

I need to simplify the right-hand side, and then see where that leads. I'll expand the left-hand side, and then solve. I'll expand and simplify on the right-hand side, and then solve. Share This Page. Terms of Use Privacy Contact. Advertising Linking to PM Site licencing. Visit Our Profiles.

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