Question

4x^2+x-33=0, y^2-10y-24=0, p^2+p-90=0

Answer 1

$1)\; \; 4x^2+x-33=0\\\\x^2+\frac{1}{4}x-\frac{33}{4}=0\\\\\Big (x^2+2\cdot \frac{1}{8}x+(\frac{1}{8})^2\Big )-(\frac{1}{8})^2-\frac{33}{4}=0\\\\ (x+\frac{1}{8})^2=\frac{529}{64}\; \; \; \Rightarrow \; \; \; x+\frac{1}{8}=\pm \frac{23}{8}\\\\a)\; \; x+\frac{1}{8}=-\frac{23}{8}\; ,\; \; x=-3\\\\b)\; \; x+\frac{1}{8}=\frac{23}{8}\; ,\; \; x=\frac{22}{8}=\frac{11}{4}$

$2)\; \; y^2-10y-24=0\\\\(y^2-2\cdot 5y+5^2)-5^2-24=0\\\\(y-5)^2=49\; \; \; \Rightarrow \; \; \; y-5=\pm 7\\\\a)\; \; y-5=-7\; ,\; \; x\y=-2\\\\b)\; \; y-5=7\; ,\; \; y=12$

$3)\; \; p^2+p-90=0\\\\\Big (p^2+2\cdot \frac{1}{2}p+(\frac{1}{2})^2\Big )-(\frac{1}{2})^2-90=0\\\\ (p+\frac{1}{2})^2=\frac{361}{4}\\\\a)\; \; p+\frac{1}{2}=-\frac{19}{2}\; ,\; \; p=-\frac{20}{2}=-10\\\\b)\; \; p+\frac{1}{2}=\frac{19}{2}\; ,\; \; p=\frac{18}{2}=9\\\\\\\star \; \; x^2\pm px+q=(x\pm \frac{p}{2})^2-(\frac{p}{2})^2+q\; \; \star$

Answer 2

4x²+x-33=0

D = 1 + 528 = 529

x₁ = (-1 + 23)/8 = 22/8 = 2,75

x₂ = (-1-23)/8 = -3

Ответ: -3; 2,75

y²-10y-24=0

х₁ = 12

х₂ = -2, по теореме Виета

Ответ: -2; 12

p²+p-90=0

D = 1 + 360 = 361

x₁ = (-1 + 19)/2 = 9

x₂ = (-1-19)/2 = -10

Ответ: -10;9

Answer 3

To find the solutions for each equation, we can use the quadratic formula:

For a quadratic equation of the form ax^2 + bx + c = 0, the solutions are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Let's solve each equation one by one:

1. 4x^2 + x - 33 = 0 a = 4, b = 1, c = -33 x = (-(1) ± √(1^2 - 4 * 4 * -33)) / 2 * 4 x = (-1 ± √(1 + 528)) / 8 x = (-1 ± √529) / 8 x = (-1 ± 23) / 8

So, the two solutions for this equation are: x1 = (23 - 1) / 8 = 22 / 8 = 11 / 4 x2 = (-23 - 1) / 8 = -24 / 8 = -3

2. y^2 - 10y - 24 = 0 a = 1, b = -10, c = -24 y = (-(-10) ± √((-10)^2 - 4 * 1 * -24)) / 2 * 1 y = (10 ± √(100 + 96)) / 2 y = (10 ± √196) / 2 y = (10 ± 14) / 2

So, the two solutions for this equation are: y1 = (10 + 14) / 2 = 24 / 2 = 12 y2 = (10 - 14) / 2 = -4 / 2 = -2

3. p^2 + p - 90 = 0 a = 1, b = 1, c = -90 p = (-(1) ± √(1^2 - 4 * 1 * -90)) / 2 * 1 p = (-1 ± √(1 + 360)) / 2 p = (-1 ± √361) / 2 p = (-1 ± 19) / 2

So, the two solutions for this equation are: p1 = (19 - 1) / 2 = 18 / 2 = 9 p2 = (-19 - 1) / 2 = -20 / 2 = -10

Final solutions for each equation:

1. x = 11/4 or x = -3
2. y = 12 or y = -2
3. p = 9 or p = -10