To calculate: The solution set of the polynomial equation 4x2−6x2−20x+30=0.

Question

To calculate: The solution set of the polynomial equation  4x2−6x2−20x+30=0.

Coldst · Accepted Answer

Formula used:
According to zero product property if,
mn=0,
Then,
m=0
Or,
n=0
Calculation:
This is a polynomial equation with one side equal to zero.

4 x^{2} - 6 x^{2} - 20 x + 30 = 0 .

Do the factors of the given equation,

2 x^{2} (2 x - 3) - 10 (2 x - 3) = 0

(2 x - 3) (2 x^{2} - 10) = 0

Set each factor equal to zero.
(2x-3)=0

x = \frac{3}{2}

Or

(2 x^{2} - 10) = 0

x^{2} = \frac{10}{2}

x = \pm \sqrt{\frac{10}{2}}

x = \pm \sqrt{5}

Check at

x = \frac{3}{2}

4 {(\frac{3}{2})}^{3} - 6 {(\frac{3}{2})}^{2} - 20 (\frac{3}{2}) + 30 \overset{?}{=} 0

\frac{54}{4} - \frac{54}{4} - 30 + 30 \overset{?}{=} 9 [0]

0=0
The left side value is equal to the right-side value. Thus, the value

x = \frac{3}{2}

is verified.
Check at

x = \sqrt{5}

4 {(\sqrt{5})}^{3} - 6 {(\sqrt{5})}^{2} - 20 (\sqrt{5}) + 30 \overset{?}{=} 0

4 \cdot 5 \sqrt{5} - 6 \cdot 5 - 20 \sqrt{5} + 30 \overset{?}{=} 0

20 \sqrt{5} - 30 - 20 \sqrt{5} + 30 \overset{?}{=} 0

0=0
The left side value is equal to the right-side value. Thus, the value

x = \sqrt{5}

is verified.
Check at

x = - \sqrt{5}

4 {(- \sqrt{5})}^{3} - 6 {(- \sqrt{5})}^{2} - 20 (- \sqrt{5}) + 30 \overset{?}{=} 0

- 4 \cdot 5 \sqrt{5} - 6 \cdot 5 + 20 \sqrt{5} + 30 \overset{?}{=} 0

- 20 \sqrt{5} - 30 + 20 \sqrt{5} + 30 \overset{?}{=} 0

0=0
The left side value is equal to the right-side value. Thus, the value

x = - \sqrt{5}

is verified.
Therefore, all together the solution set of polynomial equation

4 x^{3} - 6 x^{2} - 20 x + 30 = 0 i s {\frac{3}{2}, \pm \sqrt{5}} .

To calculate: The solution set of the polynomial equation 4x^{2