Solve (4x+6)^2=2x+3

$$( 4 x + 6 ) ^ { 2 } = 2 x + 3$$

Solve for x

$$x = -\frac{11}{8} = -1\frac{3}{8} = -1.375$$

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+6\right)^{2}.

$$16x^{2}+48x+36=2x+3$$

Subtract 2x from both sides.

$$16x^{2}+48x+36-2x=3$$

Combine 48x and -2x to get 46x.

$$16x^{2}+46x+36=3$$

Subtract 3 from both sides.

$$16x^{2}+46x+36-3=0$$

Subtract 3 from 36 to get 33.

$$16x^{2}+46x+33=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 16x^{2}+ax+bx+33. To find a and b, set up a system to be solved.

$$a+b=46 ab=16\times 33=528$$

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 528.

$$1,528 2,264 3,176 4,132 6,88 8,66 11,48 12,44 16,33 22,24$$

Calculate the sum for each pair.

$$1+528=529 2+264=266 3+176=179 4+132=136 6+88=94 8+66=74 11+48=59 12+44=56 16+33=49 22+24=46$$

The solution is the pair that gives sum 46.

$$a=22 b=24$$

Rewrite 16x^{2}+46x+33 as \left(16x^{2}+22x\right)+\left(24x+33\right).

$$\left(16x^{2}+22x\right)+\left(24x+33\right)$$

Factor out 2x in the first and 3 in the second group.

$$2x\left(8x+11\right)+3\left(8x+11\right)$$

Factor out common term 8x+11 by using distributive property.

$$\left(8x+11\right)\left(2x+3\right)$$

To find equation solutions, solve 8x+11=0 and 2x+3=0.

$$x=-\frac{11}{8} x=-\frac{3}{2}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+6\right)^{2}.

$$16x^{2}+48x+36=2x+3$$

Subtract 2x from both sides.

$$16x^{2}+48x+36-2x=3$$

Combine 48x and -2x to get 46x.

$$16x^{2}+46x+36=3$$

Subtract 3 from both sides.

$$16x^{2}+46x+36-3=0$$

Subtract 3 from 36 to get 33.

$$16x^{2}+46x+33=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, 46 for b, and 33 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-46±\sqrt{46^{2}-4\times 16\times 33}}{2\times 16}$$

Square 46.

$$x=\frac{-46±\sqrt{2116-4\times 16\times 33}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-46±\sqrt{2116-64\times 33}}{2\times 16}$$

Multiply -64 times 33.

$$x=\frac{-46±\sqrt{2116-2112}}{2\times 16}$$

Add 2116 to -2112.

$$x=\frac{-46±\sqrt{4}}{2\times 16}$$

Take the square root of 4.

$$x=\frac{-46±2}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{-46±2}{32}$$

Now solve the equation x=\frac{-46±2}{32} when ± is plus. Add -46 to 2.

$$x=-\frac{44}{32}$$

Reduce the fraction \frac{-44}{32} to lowest terms by extracting and canceling out 4.

$$x=-\frac{11}{8}$$

Now solve the equation x=\frac{-46±2}{32} when ± is minus. Subtract 2 from -46.

$$x=-\frac{48}{32}$$

Reduce the fraction \frac{-48}{32} to lowest terms by extracting and canceling out 16.

$$x=-\frac{3}{2}$$

The equation is now solved.

$$x=-\frac{11}{8} x=-\frac{3}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4x+6\right)^{2}.

$$16x^{2}+48x+36=2x+3$$

Subtract 2x from both sides.

$$16x^{2}+48x+36-2x=3$$

Combine 48x and -2x to get 46x.

$$16x^{2}+46x+36=3$$

Subtract 36 from both sides.

$$16x^{2}+46x=3-36$$

Subtract 36 from 3 to get -33.

$$16x^{2}+46x=-33$$

Divide both sides by 16.

$$\frac{16x^{2}+46x}{16}=-\frac{33}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$x^{2}+\frac{46}{16}x=-\frac{33}{16}$$

Reduce the fraction \frac{46}{16} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{23}{8}x=-\frac{33}{16}$$

Divide \frac{23}{8}, the coefficient of the x term, by 2 to get \frac{23}{16}. Then add the square of \frac{23}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{23}{8}x+\left(\frac{23}{16}\right)^{2}=-\frac{33}{16}+\left(\frac{23}{16}\right)^{2}$$

Square \frac{23}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{23}{8}x+\frac{529}{256}=-\frac{33}{16}+\frac{529}{256}$$

Add -\frac{33}{16} to \frac{529}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{23}{8}x+\frac{529}{256}=\frac{1}{256}$$

Factor x^{2}+\frac{23}{8}x+\frac{529}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{23}{16}\right)^{2}=\frac{1}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{23}{16}\right)^{2}}=\sqrt{\frac{1}{256}}$$

Simplify.

$$x+\frac{23}{16}=\frac{1}{16} x+\frac{23}{16}=-\frac{1}{16}$$

Subtract \frac{23}{16} from both sides of the equation.

$$x=-\frac{11}{8} x=-\frac{3}{2}$$