Solve for x: 24x^2 + 3x = 8x + 169x^2 - 4

Steps to Solve

1. Rewrite the equation

Start with the given equation: $$24x^2 + 3x = 8x + 169x^2 - 4$$

2. Move all terms to one side

Subtract $8x$ and $169x^2$ from both sides and add $4$ to both sides to set the equation to zero: $$24x^2 + 3x - 8x - 169x^2 + 4 = 0$$

3. Combine like terms

Combine the $x^2$ terms and the $x$ terms: $$(24x^2 - 169x^2) + (3x - 8x) + 4 = 0$$ $$-145x^2 - 5x + 4 = 0$$

4. Multiply by -1

Multiply the equation by $-1$ to make the leading coefficient positive: $$145x^2 + 5x - 4 = 0$$

5. Use the quadratic formula

The quadratic formula is given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ In this case, $a = 145$, $b = 5$, and $c = -4$.

6. Plug the values into the quadratic formula $$x = \frac{-5 \pm \sqrt{5^2 - 4(145)(-4)}}{2(145)}$$ $$x = \frac{-5 \pm \sqrt{25 + 2320}}{290}$$ $$x = \frac{-5 \pm \sqrt{2345}}{290}$$

7. Simplify the square root Since 2345 = 5 * 469 = 5 * 7 * 67, we cannot simplify $\sqrt{2345}$ further.

8. Write out the two solutions Thus the two solutions are: $$x = \frac{-5 + \sqrt{2345}}{290} \approx 0.11$$ $$x = \frac{-5 - \sqrt{2345}}{290} \approx -0.14$$