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\left(\sqrt{15x+18}\right)^{2}=\left(5x\right)^{2}
Square both sides of the equation.
15x+18=\left(5x\right)^{2}
Calculate \sqrt{15x+18} to the power of 2 and get 15x+18.
15x+18=5^{2}x^{2}
Expand \left(5x\right)^{2}.
15x+18=25x^{2}
Calculate 5 to the power of 2 and get 25.
15x+18-25x^{2}=0
Subtract 25x^{2} from both sides.
-25x^{2}+15x+18=0
Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.
a+b=15 ab=-25\times 18=-450
To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -25x^{2}+ax+bx+18. To find a and b, set up a system to be solved.
-1,450 -2,225 -3,150 -5,90 -6,75 -9,50 -10,45 -15,30 -18,25
Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -450.
-1+450=449 -2+225=223 -3+150=147 -5+90=85 -6+75=69 -9+50=41 -10+45=35 -15+30=15 -18+25=7
Calculate the sum for each pair.
a=30 b=-15
The solution is the pair that gives sum 15.
\left(-25x^{2}+30x\right)+\left(-15x+18\right)
Rewrite -25x^{2}+15x+18 as \left(-25x^{2}+30x\right)+\left(-15x+18\right).
-5x\left(5x-6\right)-3\left(5x-6\right)
Factor out -5x in the first and -3 in the second group.
\left(5x-6\right)\left(-5x-3\right)
Factor out common term 5x-6 by using distributive property.
x=\frac{6}{5} x=-\frac{3}{5}
To find equation solutions, solve 5x-6=0 and -5x-3=0.
\sqrt{15\times \frac{6}{5}+18}=5\times \frac{6}{5}
Substitute \frac{6}{5} for x in the equation \sqrt{15x+18}=5x.
6=6
Simplify. The value x=\frac{6}{5} satisfies the equation.
\sqrt{15\left(-\frac{3}{5}\right)+18}=5\left(-\frac{3}{5}\right)
Substitute -\frac{3}{5} for x in the equation \sqrt{15x+18}=5x.
3=-3
Simplify. The value x=-\frac{3}{5} does not satisfy the equation because the left and the right hand side have opposite signs.
x=\frac{6}{5}
Equation \sqrt{15x+18}=5x has a unique solution.