Question

solve each of the quadratic equations.
3x = 0.5x²
○ x = -6 or x = 0
○ x = -4 or x = 3
○ x = -2 or x = 1.5
○ x = 0 or x = 6
done
0 = 5x² - 2x + 6
○ $x=\frac{1\pm 3i}{2}$
○ $x=\frac{1\pm \sqrt{11}}{5}$
○ $x=\frac{1\pm i\sqrt{29}}{5}$
done

Explanation:

Step1: Rearrange first equation

$0.5x^2 - 3x = 0$

Step2: Factor out common term

$x(0.5x - 3) = 0$

Step3: Solve for x

$x=0$ or $0.5x-3=0 \implies x=6$

Step4: Identify quadratic for second equation

For $5x^2 -2x +6=0$, $a=5, b=-2, c=6$

Step5: Apply quadratic formula

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{2\pm\sqrt{(-2)^2-4(5)(6)}}{2(5)}$

Step6: Compute discriminant

$\sqrt{4-120}=\sqrt{-116}=\sqrt{4\times(-29)}=2i\sqrt{29}$

Step7: Simplify solution

$x=\frac{2\pm2i\sqrt{29}}{10}=\frac{1\pm i\sqrt{29}}{5}$

Answer:

For $3x=0.5x^2$: $x = 0$ or $x = 6$
For $0=5x^2 -2x +6$: $x=\frac{1\pm i\sqrt{29}}{5}$