How do you solve # sqrt (9x+1) + 6= 2x#?

Answer 1

Rewrite this equation as a quadratic and proceed to solve for #x#.

Before doing anything else, isolate the radical term on one side of the equation.

To do that, add #-6# to both sides of the equation
#sqrt(9x + 1) + cancel(6) - cancel(6) = 2x - 6#
#sqrt(9x + 1) = 2(x-3)#

Now square both sides of the equation to get rid of the radical term

#(sqrt(9x + 1))^2 = [2(x-3)]^2#
#9x + 1 = 2^2 * (x-3)^2#
#9x + 1 = 4 * (x^2 - 6x + 9)#
#9x + 1 = 4x^2 -24x + 36#

Move all your terms to one side of the equation to get

#4x^2 - 33x + 35 = 0#

Use the quadratic formula to determine the two solutions ofr this equation

#x_(1,2) = (33 +- sqrt(1089 - 4 * 4 * 35))/8#
#x_(1,2) = (33 +- sqrt(529))/8 = (33 +- 23)/8 => {(x_1 = (33 + 23)/8 = color(blue)(7)), (x_2 = (33 - 23)/8 = color(orange)(5/4)) :}#

Check to see if both solutions satisfy the original equation. Since both solutions are positive, the radical term will always be positive, so all you really need to check is whether or not the right side of the equation is positive as well.

For #x_1# you have
#2 * color(blue)(7) - 6>0#
#8 > 0 -> x_1# is #color(green)("valid")#
For #x_2# you have
#2 * color(orange)(5/4) - 6>0#
#-7/2 cancel(>) 0 -> x_2# is #color(red)("not valid")#
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Answer 2

To solve the equation sqrt(9x+1) + 6 = 2x, you can follow these steps:

  1. Start by isolating the square root term. Subtract 6 from both sides of the equation: sqrt(9x+1) = 2x - 6

  2. Square both sides of the equation to eliminate the square root: (sqrt(9x+1))^2 = (2x - 6)^2 9x + 1 = 4x^2 - 24x + 36

  3. Rearrange the equation to form a quadratic equation: 4x^2 - 24x + 36 - 9x - 1 = 0 4x^2 - 33x + 35 = 0

  4. Solve the quadratic equation. You can use factoring, completing the square, or the quadratic formula. In this case, the quadratic equation does not factor easily, so we'll use the quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / (2a)

    Plugging in the values from our equation: x = (-(-33) ± sqrt((-33)^2 - 4(4)(35))) / (2(4)) x = (33 ± sqrt(1089 - 560)) / 8 x = (33 ± sqrt(529)) / 8 x = (33 ± 23) / 8

  5. Simplify the solutions: x1 = (33 + 23) / 8 = 56 / 8 = 7/1 = 7 x2 = (33 - 23) / 8 = 10 / 8 = 5/4 = 1.25

Therefore, the solutions to the equation sqrt(9x+1) + 6 = 2x are x = 7 and x = 1.25.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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