Why can't real numbers satisfy this absolute value equation?

In summary, the absolute value equation |x^2 + 4x| = -12 has no real number solutions because the absolute value of a real number is always non-negative. Even for complex numbers, the absolute value is defined as the square root of the sum of the squares of the real and imaginary parts, which is also always non-negative. Therefore, there are no complex solutions either. This concept may be taught in advanced math classes.
  • #1
mathdad
1,283
1
Explain, in your own words, why there are no real numbers that satisfy the absolute value equation | x^2 + 4x | = - 12.

Can we say there is no real number solution here? If so, is the answer then imaginary taught in some advanced math class?
 
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  • #2
There's no solution, real or otherwise. What's the lowest value $|x^2+4x|$ can have? After answering that, consider the RHS of the equation.
 
  • #3
For x a real number, |x| is defined as "x is x is non-negative, -x if x is negative". From that it should be clear that |x| is never negative.

For x a complex number, written as a+ bi, |x| is \(\displaystyle \sqrt{a^2+ b^2}\). That is also never negative.
 
  • #4
Thank you. Someone told me that there complex solutions but that did not make sense. So, I decided to ask the real math guys. Thank you again.
 

FAQ: Why can't real numbers satisfy this absolute value equation?

What is an absolute value equation?

An absolute value equation is an equation that contains an absolute value expression, which is written as |x|, and represents the distance of a number from 0 on a number line. It can also be thought of as the positive value of a number, regardless of its sign.

How do you solve an absolute value equation?

To solve an absolute value equation, you must isolate the absolute value expression on one side of the equation. Then, you must set the expression inside the absolute value bars equal to both the positive and negative value of the other side of the equation. This will result in two equations, which can be solved to find the values of x.

Can an absolute value equation have more than one solution?

Yes, an absolute value equation can have more than one solution. This is because the absolute value of a number can be equal to both the positive and negative value of that number. Therefore, when solving an absolute value equation, you may end up with two solutions.

What is the difference between an absolute value equation and an absolute value inequality?

An absolute value equation is an equation that contains an absolute value expression, while an absolute value inequality is an inequality that contains an absolute value expression. The main difference is that when solving an absolute value equation, you will get a specific value for x, while when solving an absolute value inequality, you will get a range of values for x.

How can absolute value equations be used in real life?

Absolute value equations can be used in real life to solve problems involving distances, such as finding the distance between two points on a map or calculating the speed of a moving object. They can also be used in physics and engineering to solve problems involving magnitude and direction.

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