Solve 5x+7sqrt{x}-6=0

$$5 x + 7 \sqrt { x } - 6 = 0$$

Solve for x

$$x=\frac{9}{25}=0.36$$

Solution Steps

Subtract 5x-6 from both sides of the equation.

$$7\sqrt{x}=-\left(5x-6\right)$$

To find the opposite of 5x-6, find the opposite of each term.

$$7\sqrt{x}=-5x-\left(-6\right)$$

The opposite of -6 is 6.

$$7\sqrt{x}=-5x+6$$

Square both sides of the equation.

$$\left(7\sqrt{x}\right)^{2}=\left(-5x+6\right)^{2}$$

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

$$7^{2}\left(\sqrt{x}\right)^{2}=\left(-5x+6\right)^{2}$$

Calculate 7 to the power of 2 and get 49.

$$49\left(\sqrt{x}\right)^{2}=\left(-5x+6\right)^{2}$$

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

$$49x=\left(-5x+6\right)^{2}$$

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

$$49x=25x^{2}-60x+36$$

Subtract 25x^{2} from both sides.

$$49x-25x^{2}=-60x+36$$

Add 60x to both sides.

$$49x-25x^{2}+60x=36$$

Combine 49x and 60x to get 109x.

$$109x-25x^{2}=36$$

Subtract 36 from both sides.

$$109x-25x^{2}-36=0$$

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

$$-25x^{2}+109x-36=0$$

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.

$$a+b=109 ab=-25\left(-36\right)=900$$

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 109.

$$a=100 b=9$$

Rewrite -25x^{2}+109x-36 as \left(-25x^{2}+100x\right)+\left(9x-36\right).

$$\left(-25x^{2}+100x\right)+\left(9x-36\right)$$

Factor out 25x in the first and -9 in the second group.

$$25x\left(-x+4\right)-9\left(-x+4\right)$$

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

$$\left(-x+4\right)\left(25x-9\right)$$

To find equation solutions, solve -x+4=0 and 25x-9=0.

$$x=4 x=\frac{9}{25}$$

Substitute 4 for x in the equation 5x+7\sqrt{x}-6=0.

$$5\times 4+7\sqrt{4}-6=0$$

Simplify. The value x=4 does not satisfy the equation.

$$28=0$$

Substitute \frac{9}{25} for x in the equation 5x+7\sqrt{x}-6=0.

$$5\times \frac{9}{25}+7\sqrt{\frac{9}{25}}-6=0$$

Simplify. The value x=\frac{9}{25} satisfies the equation.

$$0=0$$

Equation 7\sqrt{x}=6-5x has a unique solution.

$$x=\frac{9}{25}$$