Solve 5x=sqrt{10+15x}

$$5 x = \sqrt { 10 + 15 x }$$

Solve for x

$$x=1$$

Solution Steps

Square both sides of the equation.

$$\left(5x\right)^{2}=\left(\sqrt{10+15x}\right)^{2}$$

Expand \left(5x\right)^{2}.

$$5^{2}x^{2}=\left(\sqrt{10+15x}\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}=\left(\sqrt{10+15x}\right)^{2}$$

Calculate \sqrt{10+15x} to the power of 2 and get 10+15x.

$$25x^{2}=10+15x$$

Subtract 10 from both sides.

$$25x^{2}-10=15x$$

Subtract 15x from both sides.

$$25x^{2}-10-15x=0$$

Divide both sides by 5.

$$5x^{2}-2-3x=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$5x^{2}-3x-2=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 5x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-3 ab=5\left(-2\right)=-10$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -10.

$$1,-10 2,-5$$

Calculate the sum for each pair.

$$1-10=-9 2-5=-3$$

The solution is the pair that gives sum -3.

$$a=-5 b=2$$

Rewrite 5x^{2}-3x-2 as \left(5x^{2}-5x\right)+\left(2x-2\right).

$$\left(5x^{2}-5x\right)+\left(2x-2\right)$$

Factor out 5x in the first and 2 in the second group.

$$5x\left(x-1\right)+2\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(5x+2\right)$$

To find equation solutions, solve x-1=0 and 5x+2=0.

$$x=1 x=-\frac{2}{5}$$

Substitute 1 for x in the equation 5x=\sqrt{10+15x}.

$$5\times 1=\sqrt{10+15\times 1}$$

Simplify. The value x=1 satisfies the equation.

$$5=5$$

Substitute -\frac{2}{5} for x in the equation 5x=\sqrt{10+15x}.

$$5\left(-\frac{2}{5}\right)=\sqrt{10+15\left(-\frac{2}{5}\right)}$$

Simplify. The value x=-\frac{2}{5} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-2=2$$

Equation 5x=\sqrt{15x+10} has a unique solution.

$$x=1$$