Question

Last question. 50 points!​

Answer 1

Answer:

The equation has two solutions which are 2 and -1.5.

Step-by-step explanation:

$\small \sf \: \sqrt{2w {}^{2} - 19w + 31} + 2 \: = 7 - 2w$

Step 1 :- Move constant to the left-hand side and change their sign.

$\small \sf ➻ \: \sqrt{2w {}^{2} - 19w + 31} \: = 7 - 2w - 2$

Step 2 :- Subtract 2 from 7.

$\small \sf ➻\: \sqrt{2w {}^{2} - 19w + 31} \: = 5- 2w$

Step 3 :- Square root on both side of expression.

$\small \sf ➻ \: ( \sqrt{2w {}^{2} - 19w + 31} )² \: = (5- 2w)²$

➻ 2w² - 19w + 31 = 25 - 20w + 4w²

Step 3 :- Move expression to the left-hand side and change their sign.

➻ 2w² - 19w + 31 - 25 + 20w - 4w² = 0

Step 4 :- Combine like terms.

➻ -2w² + w + 6 = 0

Step 5 :- Change the sign on both side of equation.

➻ 2w² - w - 6 = 0

Step 6 :- Split the term -w as 3w - 4w.

➻ 2w² + 3w - 4w - 6 = 0

Step 7 :- Factor out w from the first pair and -2 from the second pair.

➻ w ( 2w + 3 ) - 2( 2w + 3 ) = 0

Step 8 :- Factor out w + 3 from the expression.

➻ ( 2w + 3 ) ( w - 2 ) = 0

Step 9 :- Using the zero product property.

➻ 2w + 3 = 0, w - 2 = 0

Step 10 :- Solve for w.

w = $\frac{-3}{2}$ = 1.5 , and w = 2

Thus, This equation has two solutions.

Answer 2

w=-1.5

w=2

Answer:

Solution given:

$\sqrt{2w²-19w+31}+2=7-2w$

again

keep square root alone

$\sqrt{2w²-19w+31}=7-2w-2$

solve subtraction of 7-2

$\sqrt{2w²-19w+31}=5-2w$

Squaring on both side

$(\sqrt{2w²-19w+31})²=(5-2w)²$

2w²-19w+31=5²-2*5*2w+4w²

take terms one side

2w²-19w+31-25+20w-4w²=0

-2w²+w+6=0

2w²-w-6=0

doing middle term factorisation

2w²-(4-3)w-6=0

2w²-4w+3w-6=0

take common from each two term

2w(w-2)+3(w-2)=0

(w-2)(2w+3)=0

either

w=2

or

W=-3/2=-1.5