Solve the Quadratic Inequality: -x² - 25 < 0

Q: Why is the answer 'all values of x' instead of a specific range?

Because $ x^2 $ is always non-negative (never negative), and any non-negative number is automatically greater than -25. So every real number satisfies this inequality!

Q: What happens when I multiply both sides by -1?

Always flip the inequality sign when multiplying or dividing by a negative number! $ -x^2 becomes \( x^2 > -25 $.

Q: How can I check that all values work?

Pick any number and substitute it in. Try $ x = 3 $: $ -3^2 - 25 = -9 - 25 = -34 ✓. Try \( x = -10 $: \( -(-10)^2 - 25 = -100 - 25 = -125 ✓

Q: Could this inequality ever have no solution?

Not this one! Since $ -x^2 - 25 $ is always negative (because $ -x^2 \leq 0 $ and we subtract 25), the inequality \( -x^2 - 25 is always true.

Q: What if the problem was asking for when it equals zero?

Then there would be no real solutions! The equation $ -x^2 - 25 = 0 $ gives $ x^2 = -25 $, but squares of real numbers can't be negative.

Quadratic Inequalities with Negative Coefficients

Solve the following equation:

$-x^2-25<0$

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Understand the problem

Solve the following equation:

$-x^2-25<0$

Step-by-step solution

To solve the inequality $-x^2 - 25 < 0$ , we start by simplifying it. Rearrange the inequality:

$-x^2 < 25$

Multiplying through by $-1$ (reversing the inequality sign), we have:

$x^2 > -25$

Since $x^2$ is always non-negative for all real numbers (i.e., $x^2 \geq 0$ ), the smallest value $x^2$ can take is $0$ . Therefore, $x^2$ is always greater than $-25$ , since any non-negative number is greater than a negative number. Thus, this inequality holds true for all real values of $x$ .

Therefore, the solution to the inequality $-x^2 - 25 < 0$ is

All values of $x$ .

Final Answer

All values of $x$

Key Points to Remember

Essential concepts to master this topic

Rule: $x^2$ is always non-negative for all real numbers
Technique: Multiply by -1 to get $x^2 > -25$ , which is always true
Check: Test any value: $x = 0$ gives $-0 - 25 = -25 < 0$ ✓

Common Mistakes

Avoid these frequent errors

Solving as if it were an equation instead of inequality
Don't try to find specific x-values by setting $-x^2 - 25 = 0$ = no real solutions exist! This leads to thinking there's no answer. Always remember that $x^2 \geq 0$ for all real numbers, making $x^2 > -25$ true everywhere.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ x^2+4>0 $$

FAQ

Everything you need to know about this question

Why is the answer 'all values of x' instead of a specific range?

Because $x^2$ is always non-negative (never negative), and any non-negative number is automatically greater than -25. So every real number satisfies this inequality!

What happens when I multiply both sides by -1?

Always flip the inequality sign when multiplying or dividing by a negative number! $-x^2 < 25$ becomes $x^2 > -25$ .

How can I check that all values work?

Pick any number and substitute it in. Try $x = 3$ : $-3^2 - 25 = -9 - 25 = -34 < 0$ ✓. Try $x = -10$ : $-(-10)^2 - 25 = -100 - 25 = -125 < 0$ ✓

Could this inequality ever have no solution?

Not this one! Since $-x^2 - 25$ is always negative (because $-x^2 \leq 0$ and we subtract 25), the inequality $-x^2 - 25 < 0$ is always true.

What if the problem was asking for when it equals zero?

Then there would be no real solutions! The equation $-x^2 - 25 = 0$ gives $x^2 = -25$ , but squares of real numbers can't be negative.

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