Solve 2x+5sqrt{x}-3=0

$$2 x + 5 \sqrt { x } - 3 = 0$$

Solve for x

$$x=\frac{1}{4}=0.25$$

Solution Steps

Subtract 2x-3 from both sides of the equation.

$$5\sqrt{x}=-\left(2x-3\right)$$

To find the opposite of 2x-3, find the opposite of each term.

$$5\sqrt{x}=-2x-\left(-3\right)$$

The opposite of -3 is 3.

$$5\sqrt{x}=-2x+3$$

Square both sides of the equation.

$$\left(5\sqrt{x}\right)^{2}=\left(-2x+3\right)^{2}$$

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

$$5^{2}\left(\sqrt{x}\right)^{2}=\left(-2x+3\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$25\left(\sqrt{x}\right)^{2}=\left(-2x+3\right)^{2}$$

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

$$25x=\left(-2x+3\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-2x+3\right)^{2}.

$$25x=4x^{2}-12x+9$$

Subtract 4x^{2} from both sides.

$$25x-4x^{2}=-12x+9$$

Add 12x to both sides.

$$25x-4x^{2}+12x=9$$

Combine 25x and 12x to get 37x.

$$37x-4x^{2}=9$$

Subtract 9 from both sides.

$$37x-4x^{2}-9=0$$

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

$$-4x^{2}+37x-9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=37 ab=-4\left(-9\right)=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Calculate the sum for each pair.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

The solution is the pair that gives sum 37.

$$a=36 b=1$$

Rewrite -4x^{2}+37x-9 as \left(-4x^{2}+36x\right)+\left(x-9\right).

$$\left(-4x^{2}+36x\right)+\left(x-9\right)$$

Factor out 4x in the first and -1 in the second group.

$$4x\left(-x+9\right)-\left(-x+9\right)$$

Factor out common term -x+9 by using distributive property.

$$\left(-x+9\right)\left(4x-1\right)$$

To find equation solutions, solve -x+9=0 and 4x-1=0.

$$x=9 x=\frac{1}{4}$$

Substitute 9 for x in the equation 2x+5\sqrt{x}-3=0.

$$2\times 9+5\sqrt{9}-3=0$$

Simplify. The value x=9 does not satisfy the equation.

$$30=0$$

Substitute \frac{1}{4} for x in the equation 2x+5\sqrt{x}-3=0.

$$2\times \frac{1}{4}+5\sqrt{\frac{1}{4}}-3=0$$

Simplify. The value x=\frac{1}{4} satisfies the equation.

$$0=0$$

Equation 5\sqrt{x}=3-2x has a unique solution.

$$x=\frac{1}{4}$$