Question

Help me with this two I don't understand

Answer 1

Explanation:

5.

`(5 + 4 √(7) ){x}^(2) + (4 - 2 √(7) ) x- 1 = 0`

Simplify both radicals.

`(5 + \sqrt{112) {x}^(2) } + (4 - √(28) )x - 1 = 0`

Apply Quadratic Formula

First. find the discramnint.

`(4 - √(28) ) {}^(2) - 4(5 + √(112) )( - 1) = 64`

Now find the divisor 2a.

`2(5 + √(112) ) = 10 + 8 √(7)`

Then,take the square root of the discrimant.

`√(64) = 8`

Finally, add -b.

`- (4 + 2 √(7) )`

So our possible root is

`- (4 + 2 √(7) ) + (8)/(10 + 8 √(7) )`

Which simplified gives us

`( 4 + 2 √(7) )/(10 + 8 √(7) )`

Rationalize the denominator.

`(4 + 2 √(7) )/(10 + 8 √(7) ) * (10 - 8 √(7) )/(10 - 8 √(7) ) = ( - 72 - 12 √(7) )/( - 348)`

Which simplified gives us

`(6 + √(7) )/(29)`.

6. The answer is 2.

Answer 2

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Answer:

5. x = (6 +√7)/29; a=6, b=1, c=29

6. x = 2

Explanation:

5.

The quadratic formula can be used, where a=(5+4√7), b=(4-2√7), c=-1.

`x=(-b+√(b^2-4ac))/(2a)=\frac{-(4-2√(7))+\sqrt{(4-2√(7))^2-4(5+4√(7)})(-1)}{2(5+4√(7))}\\\\=\frac{-4+2√(7)+\sqrt{16-16√(7)+28+20+16√(7)}}{10+8√(7)}=(4+2√(7))/(2(5+4√(7)))\\\\=((2+√(7))(5-4√(7)))/((5+4√(7))(5-4√(7)))=(10-3√(7)-28)/(25-112)=\boxed{(6+√(7))/(29)}`

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6.

Use the substitution z=3^x to put the equation in the form ...

z² -3z -54 = 0

(z -9)(z +6) = 0 . . . . . factor

z = 9 or -6 . . . . . . . . value of z that make the factors zero

Only the positive solution is useful, since 3^x cannot be negative.

z = 9 = 3^2 = 3^x . . . . use the value of z to find x

x = 2