Solve 6x-7sqrt{x}-3=0

$$6 x - 7 \sqrt { x } - 3 = 0$$

Solve for x

$$x = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

Solution Steps

Add 3 to both sides. Anything plus zero gives itself.

$$6x-7\sqrt{x}=3$$

Subtract 6x from both sides of the equation.

$$-7\sqrt{x}=3-6x$$

Square both sides of the equation.

$$\left(-7\sqrt{x}\right)^{2}=\left(3-6x\right)^{2}$$

Expand \left(-7\sqrt{x}\right)^{2}.

$$\left(-7\right)^{2}\left(\sqrt{x}\right)^{2}=\left(3-6x\right)^{2}$$

Calculate -7 to the power of 2 and get 49.

$$49\left(\sqrt{x}\right)^{2}=\left(3-6x\right)^{2}$$

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

$$49x=\left(3-6x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3-6x\right)^{2}.

$$49x=9-36x+36x^{2}$$

Add 36x to both sides.

$$49x+36x=9+36x^{2}$$

Combine 49x and 36x to get 85x.

$$85x=9+36x^{2}$$

Subtract 36x^{2} from both sides.

$$85x-36x^{2}=9$$

Subtract 9 from both sides.

$$85x-36x^{2}-9=0$$

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

$$-36x^{2}+85x-9=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -36x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=85 ab=-36\left(-9\right)=324$$

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 324.

$$1,324 2,162 3,108 4,81 6,54 9,36 12,27 18,18$$

Calculate the sum for each pair.

$$1+324=325 2+162=164 3+108=111 4+81=85 6+54=60 9+36=45 12+27=39 18+18=36$$

The solution is the pair that gives sum 85.

$$a=81 b=4$$

Rewrite -36x^{2}+85x-9 as \left(-36x^{2}+81x\right)+\left(4x-9\right).

$$\left(-36x^{2}+81x\right)+\left(4x-9\right)$$

Factor out -9x in -36x^{2}+81x.

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

Factor out common term 4x-9 by using distributive property.

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

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

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

Substitute \frac{9}{4} for x in the equation 6x-7\sqrt{x}-3=0.

$$6\times \frac{9}{4}-7\sqrt{\frac{9}{4}}-3=0$$

Simplify. The value x=\frac{9}{4} satisfies the equation.

$$0=0$$

Substitute \frac{1}{9} for x in the equation 6x-7\sqrt{x}-3=0.

$$6\times \frac{1}{9}-7\sqrt{\frac{1}{9}}-3=0$$

Simplify. The value x=\frac{1}{9} does not satisfy the equation.

$$-\frac{14}{3}=0$$

Equation -7\sqrt{x}=3-6x has a unique solution.

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