Solve 2x + 12x - 3 = 4x^2 - 3.

Steps to Solve

1. Combine Like Terms on the Left Side

Start by combining the terms with $x$ on the left side of the equation:

$$ 2x + 12x - 3 = 4x^2 - 3 $$

This simplifies to:

$$ 14x - 3 = 4x^2 - 3 $$

2. Set the Equation to Zero

Next, we want to set the equation to zero. We can do this by moving all terms to one side:

$$ 4x^2 - 14x + 3 = 0 $$

3. Identify Coefficients for the Quadratic Formula

Now we need to identify the coefficients $a$, $b$, and $c$ for the quadratic formula, which is $ax^2 + bx + c = 0$. Here, we have:

• $a = 4$
• $b = -14$
• $c = 3$

4. Use Quadratic Formula

Next, use the quadratic formula, which is:

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

Substituting the values we found:

$$ x = \frac{-(-14) \pm \sqrt{(-14)^2 - 4 \cdot 4 \cdot 3}}{2 \cdot 4} $$

5. Simplify the Equation

Now, simplify the expression:

First, calculate discriminant:

$$ (-14)^2 = 196 $$

Now calculate $4 \cdot 4 \cdot 3 = 48$.

Thus:

$$ 196 - 48 = 148 $$

Now plug back into the formula:

$$ x = \frac{14 \pm \sqrt{148}}{8} $$

6. Simplify the Square Root

Since $\sqrt{148} = \sqrt{4 \cdot 37} = 2\sqrt{37}$, we can simplify:

$$ x = \frac{14 \pm 2\sqrt{37}}{8} $$

Now reduce the fraction:

$$ x = \frac{7 \pm \sqrt{37}}{4} $$