Solve the Quadratic Equation: 10x² - 65x + 135 = 4x² + 13x - 45

To solve this problem, we'll follow these steps:

• Step 1: Simplify the equation by gathering all terms on one side.
• Step 2: Combine like terms to express the equation in standard quadratic form.
• Step 3: Apply the quadratic formula to find the values of $x$.

Now, let's work through each step:

Step 1: Starting with the equation $10x^2 - 65x + 135 = 4x^2 + 13x - 45$, subtract $4x^2$, $13x$, and add $45$ on both sides:

$10x^2 - 65x + 135 - 4x^2 - 13x + 45 = 0$

Step 2: Combine like terms:

$(10x^2 - 4x^2) + (-65x - 13x) + (135 + 45) = 0$

This simplifies to $6x^2 - 78x + 180 = 0$.

Step 3: Identify the coefficients $a = 6$, $b = -78$, and $c = 180$. Use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substitute the values:

$x = \frac{-(-78) \pm \sqrt{(-78)^2 - 4 \cdot 6 \cdot 180}}{2 \cdot 6}$

$x = \frac{78 \pm \sqrt{6084 - 4320}}{12}$

$x = \frac{78 \pm \sqrt{1764}}{12}$

$x = \frac{78 \pm 42}{12}$

Calculating the two possible values:

$x_1 = \frac{78 + 42}{12} = \frac{120}{12} = 10$

$x_2 = \frac{78 - 42}{12} = \frac{36}{12} = 3$

Therefore, the solutions to the equation are $x_1 = 10$ and $x_2 = 3$.

The correct answer according to the provided choices is $x_1=10$ and $x_2=3$, which corresponds to choice 4.