How do you solve #x-1= sqrt( 6x+10)#?

Question

Eli Assouline · Accepted Answer

#x = 9#.

#(x - 1)^2 = (sqrt(6x + 10))^2# #x^2 - 2x + 1 = 6x + 10# #x^2 - 8x - 9 = 0# #(x - 9)(x + 1) = 0# #x = 9 and x =-1# However, since extraneous solutions are a distinct possibility, let's check our solution inside the initial equation. #9 - 1 =^? sqrt(6 xx 9 + 10)# #8 = 8" "color(green)(√)# AND #-1 - 1=^? sqrt(6 xx -1 + 10)# #-2 != 2" "color(red)(xx)# Hence, the only true solution is #x = 9#. Hopefully this helps!

Ethan Adkins · Answer

undefined To solve the equation x-1= sqrt(6x+10), we can follow these steps:

1. Start by isolating the square root term on one side of the equation. Add 1 to both sides to get x = 1 + sqrt(6x+10).

2. Square both sides of the equation to eliminate the square root. This gives us x^2 = (1 + sqrt(6x+10))^2.

3. Expand the right side of the equation using the formula (a + b)^2 = a^2 + 2ab + b^2. In this case, a = 1 and b = sqrt(6x+10). So, we have x^2 = 1^2 + 2(1)(sqrt(6x+10)) + (sqrt(6x+10))^2.

4. Simplify the equation by squaring the terms and combining like terms. This gives us x^2 = 1 + 2sqrt(6x+10) + 6x + 10.

5. Rearrange the equation to bring all terms to one side, resulting in x^2 - 6x - 2sqrt(6x+10) - 11 = 0.

6. At this point, we have a quadratic equation. To solve it, we can use factoring, completing the square, or the quadratic formula. However, factoring may not be straightforward in this case, so let's use the quadratic formula.

7. Apply the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by x = (-b ± sqrt(b^2 - 4ac)) / (2a). In our equation, a = 1, b = -6, and c = -11.

8. Substitute these values into the quadratic formula and simplify to find the solutions for x.

x = (6 ± sqrt((-6)^2 - 4(1)(-2sqrt(6x+10) - 11))) / (2(1))

9. Continue simplifying the equation by performing the necessary calculations.

x = (6 ± sqrt(36 + 8sqrt(6x+10) + 44)) / 2

10. Simplify further.

x = (6 ± sqrt(80 + 8sqrt(6x+10))) / 2

11. Simplify the square root term.

x = (6 ± sqrt(4(20 + 2sqrt(6x+10)))) / 2

12. Simplify again.

x = (6 ± 2sqrt(20 + 2sqrt(6x+10))) / 2

13. Divide both the numerator and denominator by 2.

x = 3 ± sqrt(20 + 2sqrt(6x+10))

14. At this point, we have two possible solutions: x = 3 + sqrt(20 + 2sqrt(6x+10)) and x = 3 - sqrt(20 + 2sqrt(6x+10)).

These are the solutions to the equation x-1= sqrt(6x+10).