Solve sqrt{7x+67}=2x+5

$$\sqrt { 7 x + 67 } = 2 x + 5$$

Solve for x

$$x=2$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{7x+67}\right)^{2}=\left(2x+5\right)^{2}$$

Calculate \sqrt{7x+67} to the power of 2 and get 7x+67.

$$7x+67=\left(2x+5\right)^{2}$$

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

$$7x+67=4x^{2}+20x+25$$

Subtract 4x^{2} from both sides.

$$7x+67-4x^{2}=20x+25$$

Subtract 20x from both sides.

$$7x+67-4x^{2}-20x=25$$

Combine 7x and -20x to get -13x.

$$-13x+67-4x^{2}=25$$

Subtract 25 from both sides.

$$-13x+67-4x^{2}-25=0$$

Subtract 25 from 67 to get 42.

$$-13x+42-4x^{2}=0$$

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

$$-4x^{2}-13x+42=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx+42. To find a and b, set up a system to be solved.

$$a+b=-13 ab=-4\times 42=-168$$

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 -168.

$$1,-168 2,-84 3,-56 4,-42 6,-28 7,-24 8,-21 12,-14$$

Calculate the sum for each pair.

$$1-168=-167 2-84=-82 3-56=-53 4-42=-38 6-28=-22 7-24=-17 8-21=-13 12-14=-2$$

The solution is the pair that gives sum -13.

$$a=8 b=-21$$

Rewrite -4x^{2}-13x+42 as \left(-4x^{2}+8x\right)+\left(-21x+42\right).

$$\left(-4x^{2}+8x\right)+\left(-21x+42\right)$$

Factor out 4x in the first and 21 in the second group.

$$4x\left(-x+2\right)+21\left(-x+2\right)$$

Factor out common term -x+2 by using distributive property.

$$\left(-x+2\right)\left(4x+21\right)$$

To find equation solutions, solve -x+2=0 and 4x+21=0.

$$x=2 x=-\frac{21}{4}$$

Substitute 2 for x in the equation \sqrt{7x+67}=2x+5.

$$\sqrt{7\times 2+67}=2\times 2+5$$

Simplify. The value x=2 satisfies the equation.

$$9=9$$

Substitute -\frac{21}{4} for x in the equation \sqrt{7x+67}=2x+5.

$$\sqrt{7\left(-\frac{21}{4}\right)+67}=2\left(-\frac{21}{4}\right)+5$$

Simplify. The value x=-\frac{21}{4} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{11}{2}=-\frac{11}{2}$$

Equation \sqrt{7x+67}=2x+5 has a unique solution.

$$x=2$$