Question

1. $25x^2 + 35x - 30 = 0$
2. $3x^3 - 16x^2 + 5x = 0$

Explanation:

Step1: Simplify quadratic equation (Q3)

Divide all terms by 5:
$5x^2 + 7x - 6 = 0$

Step2: Apply quadratic formula (Q3)

For $ax^2+bx+c=0$, $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. Here $a=5, b=7, c=-6$:
$x=\frac{-7\pm\sqrt{7^2-4(5)(-6)}}{2(5)}=\frac{-7\pm\sqrt{49+120}}{10}=\frac{-7\pm\sqrt{169}}{10}=\frac{-7\pm13}{10}$

Step3: Calculate roots (Q3)

First root: $x=\frac{-7+13}{10}=\frac{6}{10}=\frac{3}{5}$
Second root: $x=\frac{-7-13}{10}=\frac{-20}{10}=-2$

Step4: Factor cubic equation (Q4)

Factor out $x$:
$x(3x^2 - 16x + 5) = 0$

Step5: Factor quadratic term (Q4)

Factor $3x^2-16x+5$:
$3x^2-16x+5=(3x-1)(x-5)$
So the equation becomes $x(3x-1)(x-5)=0$

Step6: Find roots (Q4)

Set each factor to 0:
$x=0$, $3x-1=0\Rightarrow x=\frac{1}{3}$, $x-5=0\Rightarrow x=5$

Answer:

For $25x^2 + 35x - 30 = 0$: $x=-2$ and $x=\frac{3}{5}$
For $3x^3 - 16x^2 + 5x = 0$: $x=0$, $x=\frac{1}{3}$, and $x=5$