Question

решите с объяснением, пожалуйста : x-4√(x+4)-1< 0

Answer 1

X-4√(x+4)-1< 0
х-1<4√(x+4)
возведем обе части уравнения в квадрат
(х-1)²<16(х+4)
х²-2х+1<16х+64
х²-2х+1-16х-64<0
х²-18х-63<0
(x-21)(x+3)<0

х²-18х-63=0
D=324+252=576
x1=18+24/2=42/2=21
x2=18-24/2=-6/2=-3
х²-18х-63=(x-21)(x+3)

(x-21)(x+3)<0

__+__-3__-__21__+__

значит нашим решением будет интервал:
(-3; 21)

Answer 2

I can help you with your math problem. Here is the solution:

To solve the inequality $x-4\sqrt{x+4}-1<0$, we need to find the values of $x$ that make the expression negative. We can do this by following these steps:

1. Isolate the square root term by adding $1$ to both sides: $x-4\sqrt{x+4}<1$. 2. Square both sides to eliminate the square root: $(x-4\sqrt{x+4})^2<1^2$. Be careful, this may introduce extraneous solutions that do not satisfy the original inequality. 3. Expand the left side using the formula $(a-b)^2=a^2-2ab+b^2$: $x^2-8x\sqrt{x+4}+16(x+4)<1$. 4. Simplify the left side by distributing the $16$: $x^2-8x\sqrt{x+4}+16x+64<1$. 5. Subtract $1$ from both sides to get a zero on the right side: $x^2-8x\sqrt{x+4}+16x+63<0$. 6. Factor the left side using the quadratic formula or by trial and error: $(x+9)(x-7\sqrt{x+4})<0$. 7. Find the zeros of the left side by setting each factor equal to zero: $x+9=0$ or $x-7\sqrt{x+4}=0$. 8. Solve for $x$ in each case: $x=-9$ or $x=7\sqrt{x+4}$. 9. Simplify the second equation by squaring both sides: $x=-9$ or $x^2=49(x+4)$. 10. Solve the second equation by rearranging the terms and factoring: $x=-9$ or $x^2-49x-196=0$ or $(x-14)(x+14)=0$. 11. Solve for $x$ in each case: $x=-9$ or $x=14$ or $x=-14$. 12. Check which values of $x$ satisfy the original inequality by plugging them into the expression: $x-4\sqrt{x+4}-1<0$. 13. Reject any values of $x$ that make the expression undefined or positive: $x=-9$ is rejected because it makes the square root undefined, $x=14$ is rejected because it makes the expression positive. 14. The only value of $x$ that satisfies the original inequality is $x=-14$.

Therefore, the solution is $x=-14$.Ответь+подробно.+решите+с+объяснением,+пожалуйста+:+x-4√(x+4)-1<+0