finding zeros?

Question

if the equation is x squared + 4x + 8=0 and the instructions say "find the zeros," how would you do that?
also, when you are "finding the zeros" of an equation, what exactly are you doing? like, are you finding all the numbers that will make your equation equal to 0? or what?
thanks.

Answer 1

x^2 + 4x + 8 = 0

Usually, if it's an equation (equated to 0), you'd normally be asked to find the roots (which are, in essence, a solution to x).

The term "zero" is used if it were a function; in your case, the above problem is equivalent to finding the zeroes of
f(x) = x^2 + 4x + 8

In a nutshell: "zeros" are used to find a function, "roots" are used for polynomial equations.

Like you suspected, finding the "zeros" of a function mean values of x which make f(x) equal to 0.

Let's solve this.

x^2 + 4x + 8 = 0

We *could* use the quadratic formula, but instead I'm going to complete the square.

x^2 + 4x + 4 + 4 = 0
x^2 + 4x + 4 = -4
(x + 2)^2 = -4

At this point, we would take the square root of both sides; however, you cannot take the square root of -4 if you're confined to real numbers (which, I suspect, you are, given you're asking about zeros). Therefore, there is no real solution.

There are, however, complex solutions.

(x + 2)^2 = -4

Taking the square root of both sides, and we will have to add a "Plus or minus" on the right hand side (this is a typical step when taking the square root of both sides).

x + 2 = +/- sqrt(-4)

There's a term for the square root of -1, called i. Since

x + 2 = +/- sqrt ( (4) (-1) )

Splitting the square root into two,

x + 2 = +/- sqrt(4)sqrt(-1)
x + 2 = +/- 2i

And bringing the 2 over,

x = -2 +/- 2i

So your solutions for x are:
x = {-2 + 2i , -2 - 2i}

Answer 2

Sounds like you're studying polynomial functions.

In general, the following 4 statements are equivalent:
x=a is a zero of the function f
x=a is a solution of the polynomial equation f(x)=0
(x-a) is a factor of the polynomial f(x)
(a,0) is an x-intercept of the graph of f.

so with the equation you've given, since we can't factor it easily - for example (x+2)(x+4) is wrong - use a graphing calculator utility to graph the function (equation). inspect the graph to find the x-intercepts and you've got your zeroes.

by the way, if the original equation was x squared +6x +8, then the zeroes would be -2 and -4.

also, as problems progress in difficulty and you need to do powers of x higher than 2, you'll either definitely need to use the calculator, or, if the function ends in an x term, factor out the x and work on the remaining equation.
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Answer 3

If the equation is y = x^2 + 4x +8 = 0, you are trying to find the values of x that will make y = 0. These are also called the roots of the equation. It indicates that where the graph crosses the x-axis, we have a real root or zero of the equation.

Some equations may not have a graph that crosses the x-axis. In this case, there are no zeroes, but the roots are imaginary. Your example equation is a case where the roots of the equation are imaginary. The graph is entirely above the x-axis and so can never cross it to give zeroes.

Answer 4

When you are finding the zeros of an equation, you are finding all values of x such that f(x) = 0. This corresponds to the points where the graph of the function intersects the x-axis.

To find the zeros, you have two main options:

(1) If the equation factors nicely, then factor it and check to see when the factored terms are 0.

(2) If it doesn't factor nicely, then use the quadratic formula.

Answer 5

The answer to the second part of your question is yes, you are finding the values for x that make it equal zero.

Use the quadratic formula to find your zeros...(note: I use | for square root)
(-4 (+/-) |(4^2-4*1*8))/2*1
(-4 (+/-) |(16-32))/2
(-4 (+/-) |-16)/2
(-4 (+/-) 4i)/2
(-2+2i)(-2-2i) <---- These are your roots....notice that they are imaginary (not Real) numbers

Answer 6

yes...you are trying to find all the x values that make the answer zero.
One way is to graph the equation on your graphing calculator and see where the lines cross the x-axis (which is 0 on the y).
Or you can factor it and solve it or use the quadratic formula

Answer 7

P(x) has one 0 at -a million potential it could factorized into P(x) = Q(x)*(x+a million). you are able to then use long branch to locate Q(x). on the different hand, by potential of observing P(x), you additionally can see P(x)=(x^2-a million)(x^2-2)=(x-a million)(x+a million) *(x-sqrt(2))(x+sqrt(2)). So the different 3 zeros are a million, sqrt(2), and -sqrt(2). (b) slightly complicated with the help of potential of looking. yet long branch could desire to paintings.

Answer 8

when find zeroes, that means you find the value of x that makes y is equal to 0

this equation can not be factored, so you have to use quadractic fomula