Solve 5x+sqrt{x}=6

$$5 x + \sqrt { x } = 6$$

Solve for x

$$x=1$$

Solution Steps

Subtract 5x from both sides of the equation.

$$\sqrt{x}=6-5x$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(6-5x\right)^{2}$$

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

$$x=\left(6-5x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-5x\right)^{2}.

$$x=36-60x+25x^{2}$$

Subtract 36 from both sides.

$$x-36=-60x+25x^{2}$$

Add 60x to both sides.

$$x-36+60x=25x^{2}$$

Combine x and 60x to get 61x.

$$61x-36=25x^{2}$$

Subtract 25x^{2} from both sides.

$$61x-36-25x^{2}=0$$

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

$$-25x^{2}+61x-36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -25x^{2}+ax+bx-36. To find a and b, set up a system to be solved.

$$a+b=61 ab=-25\left(-36\right)=900$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 900.

$$1,900 2,450 3,300 4,225 5,180 6,150 9,100 10,90 12,75 15,60 18,50 20,45 25,36 30,30$$

Calculate the sum for each pair.

$$1+900=901 2+450=452 3+300=303 4+225=229 5+180=185 6+150=156 9+100=109 10+90=100 12+75=87 15+60=75 18+50=68 20+45=65 25+36=61 30+30=60$$

The solution is the pair that gives sum 61.

$$a=36 b=25$$

Rewrite -25x^{2}+61x-36 as \left(-25x^{2}+36x\right)+\left(25x-36\right).

$$\left(-25x^{2}+36x\right)+\left(25x-36\right)$$

Factor out -x in -25x^{2}+36x.

$$-x\left(25x-36\right)+25x-36$$

Factor out common term 25x-36 by using distributive property.

$$\left(25x-36\right)\left(-x+1\right)$$

To find equation solutions, solve 25x-36=0 and -x+1=0.

$$x=\frac{36}{25} x=1$$

Substitute \frac{36}{25} for x in the equation 5x+\sqrt{x}=6.

$$5\times \frac{36}{25}+\sqrt{\frac{36}{25}}=6$$

Simplify. The value x=\frac{36}{25} does not satisfy the equation.

$$\frac{42}{5}=6$$

Substitute 1 for x in the equation 5x+\sqrt{x}=6.

$$5\times 1+\sqrt{1}=6$$

Simplify. The value x=1 satisfies the equation.

$$6=6$$

Equation \sqrt{x}=6-5x has a unique solution.

$$x=1$$