Question

(3x^2 - 5x)^2 - (5x - 3)^2 = 0;​

Answer 1

Ответ:

{ 3 ; 1/3 ; 1 ; -1 }

Объяснение:

(3x² - 5x)² - (5x - 3)² = 0

(3x²-5x-(5x-3))(3x²-5x+(5x-3))=0

(3x²-5x-5x+3)(3x²-5x+5x-3)=0

(3x²-10x+3)(3x²-3)=0

3x²-10x+3=0

D=100-36=64

√64=8

x=(10±8)/6 = 3 ; 1/3

3x²-3=0

3x²=3

x²=1

x=±1

Answer 2

To solve the equation $(3x^2 - 5x)^2 - (5x - 3)^2 = 0$, you can use the difference of squares factorization. Here's how to do it step by step:

1. Recognize that this equation is in the form of a difference of squares, where $(a^2 - b^2)$ can be factored as $(a + b)(a - b)$.

2. Rewrite the equation as follows:

$(3x^2 - 5x + 5x - 3)(3x^2 - 5x - 5x + 3) = 0$

3. Simplify the expressions inside the parentheses:

$(3x^2 - 3)(3x^2 - 10x + 3) = 0$

4. Now you have two factors to consider:

a) $3x^2 - 3 = 3(x^2 - 1)$ b) $3x^2 - 10x + 3$

Let's solve each factor separately:

a) $3x^2 - 3 = 3(x^2 - 1)$

Now, set each factor equal to zero:

i) $3 = 0$ (This is not possible.) ii) $x^2 - 1 = 0$

Solve for $x^2$:

$x^2 = 1$

Take the square root of both sides:

$x = \pm 1$

So, for this part of the equation, you have two solutions: $x = 1$ and $x = -1$.

b) Now, let's solve the second factor: $3x^2 - 10x + 3 = 0$

You can use the quadratic formula to solve for $x$:

The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

In this case, $a = 3$, $b = -10$, and $c = 3$.

$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(3)(3)}}{2(3)}$

Simplify:

$x = \frac{10 \pm \sqrt{100 - 36}}{6}$

$x = \frac{10 \pm \sqrt{64}}{6}$

$x = \frac{10 \pm 8}{6}$

Now, calculate the two possible values for $x$:

i) $x = \frac{10 + 8}{6} = \frac{18}{6} = 3$ ii) $x = \frac{10 - 8}{6} = \frac{2}{6} = \frac{1}{3}$

So, for this part of the equation, you have two more solutions: $x = 3$ and $x = \frac{1}{3}$.

In total, the equation has four solutions:

1. $x = 1$
2. $x = -1$
3. $x = 3$
4. $x = \frac{1}{3}$

These are the solutions to the equation $(3x^2 - 5x)^2 - (5x - 3)^2 = 0$.