Trying to work this problem but not sure where to start.

Hi CiJi,

Let's begin by writing out our equation as you indicated

(the 6x^2 and -3/2 terms being under the square root sign): sqrt(6x^2) + 2x - sqrt(3/2) = 0

Now let's isolate the sqrt(6x^2) term on the left side by moving the other two terms to the right side (reversing their signs on the right side) - this will cancel out these terms on the left side: -2x cancels out +2x and plus sqrt(3/2) cancels out minus sqrt(3/2): sqrt(6x^2) = -2x + sqrt(3/2)

Now let's get rid of the radical (square root sign) on the left side by squaring both sides of the equation remembering that on the right side - squaring a binomial expression is the same as multiplying the binomial expression by itself: 6x^2 = (-2x + sqrt(3/2))(-2x + sqrt(3/2))

Now let's multiply the two binomial expressions on the right side by multiplying each term in the 1st set of parens by each term in the 2nd set of parens:

6x^2 = (-2x)(-2x) + (-2x)(sqrt(3/2) + (sqrt(3/2))(-2x) + (sqrt(3/2))(sqrt(3/2))

Now let's write out the multiplication results on the right side:

6x^2 = 4x^2 - 2sqrt(3/2)x - 2sqrt(3/2)x + 3/2

Now let's simplify the right side by combining like terms: 6x^2 = 4x^2 - 4sqrt(3/2)x + 3/2

Now let's set the right side to zero and move the terms on the right side to the left side by reversing the sign of each term to cancel out these terms on the right side:

6x^2 - 4x^2 + 4sqrt(3/2)x - 3/2 = 0

Now let's simplify the left side by combining like terms: 2x^2 + 4sqrt(3/2)x - 3/2 = 0

We now have the standard form of a quadratic equation (ax^2 + bx + c = 0):

2x^2 + 4sqrt(3/2)x - 3/2 = 0

Our coefficients (a, b, c) are: a = 2 and b = 4sqrt(3/2) and c = -3/2

We can now use the quadratic formula to solve for x: x = (-b +/- sqrt(b^2 - 4ac)) / 2a

Plugging in our coefficient values for a and b and c:

x = (-4sqrt(3/2) +/- sqrt((4sqrt(3/2))^2 - 4(2)(-3/2))) / 2(2)

Squaring b coefficient (4sqrt(3/2)^2):

x = (-4sqrt(3/2) +/- sqrt(16(3/2) - 4(2)(-3/2))) / 2(2)

Performing all multiplication operations:

x = (-4sqrt(3/2) +/- sqrt(24 + 12)) / 4

Performing addition operation under square root sign:

x = (-4sqrt(3/2) +/- sqrt(36)) / 4

Extracting perfect square (taking square root of 36):

x = (-4sqrt(3/2) +/- 6) / 4

Final simplification (dividing each term in numerator and denominator by 4 to get rid of fraction):

x = -sqrt(3/2) +/- 3/2

**Final Exact Answers for x** in radical form:

**x = -sqrt(3/2) + 3/2**

**x = -sqrt(3/2) - 3/2**

**Final Approximate Answers for x** in decimal form rounded to nearest hundredth:

**x = -sqrt(3/2) + 3/2 = -1.22 + 1.50 = 0.28**

**x = -sqrt(3/2) - 3/2 = -1.22 - 1.50 = -2.72**

Whew !!

This was definitely a challenging problem, but I trust the above stepwise explanation will be helpful to you.

Thanks for submitting your question.

Regards, Jordan.

## Comments

?? How is 6x

^{2}a square root?Is 3/2 actually 3v2, or three halves, = 1.5?