Solve (5+2x)(16+3x)=133

$$( 5 + 2 x ) ( 16 + 3 x ) = 133$$

Solve for x

$$x = -\frac{53}{6} = -8\frac{5}{6} \approx -8.833333333$$

$$x=1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5+2x by 16+3x and combine like terms.

$$80+47x+6x^{2}=133$$

Subtract 133 from both sides.

$$80+47x+6x^{2}-133=0$$

Subtract 133 from 80 to get -53.

$$-53+47x+6x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$6x^{2}+47x-53=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 47 for b, and -53 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-47±\sqrt{47^{2}-4\times 6\left(-53\right)}}{2\times 6}$$

Square 47.

$$x=\frac{-47±\sqrt{2209-4\times 6\left(-53\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-47±\sqrt{2209-24\left(-53\right)}}{2\times 6}$$

Multiply -24 times -53.

$$x=\frac{-47±\sqrt{2209+1272}}{2\times 6}$$

Add 2209 to 1272.

$$x=\frac{-47±\sqrt{3481}}{2\times 6}$$

Take the square root of 3481.

$$x=\frac{-47±59}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-47±59}{12}$$

Now solve the equation x=\frac{-47±59}{12} when ± is plus. Add -47 to 59.

$$x=\frac{12}{12}$$

Divide 12 by 12.

$$x=1$$

Now solve the equation x=\frac{-47±59}{12} when ± is minus. Subtract 59 from -47.

$$x=-\frac{106}{12}$$

Reduce the fraction \frac{-106}{12} to lowest terms by extracting and canceling out 2.

$$x=-\frac{53}{6}$$

The equation is now solved.

$$x=1 x=-\frac{53}{6}$$

Steps for Completing the Square

Use the distributive property to multiply 5+2x by 16+3x and combine like terms.

$$80+47x+6x^{2}=133$$

Subtract 80 from both sides.

$$47x+6x^{2}=133-80$$

Subtract 80 from 133 to get 53.

$$47x+6x^{2}=53$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$6x^{2}+47x=53$$

Divide both sides by 6.

$$\frac{6x^{2}+47x}{6}=\frac{53}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{47}{6}x=\frac{53}{6}$$

Divide \frac{47}{6}, the coefficient of the x term, by 2 to get \frac{47}{12}. Then add the square of \frac{47}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{47}{6}x+\left(\frac{47}{12}\right)^{2}=\frac{53}{6}+\left(\frac{47}{12}\right)^{2}$$

Square \frac{47}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{47}{6}x+\frac{2209}{144}=\frac{53}{6}+\frac{2209}{144}$$

Add \frac{53}{6} to \frac{2209}{144} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{47}{6}x+\frac{2209}{144}=\frac{3481}{144}$$

Factor x^{2}+\frac{47}{6}x+\frac{2209}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{47}{12}\right)^{2}=\frac{3481}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{47}{12}\right)^{2}}=\sqrt{\frac{3481}{144}}$$

Simplify.

$$x+\frac{47}{12}=\frac{59}{12} x+\frac{47}{12}=-\frac{59}{12}$$

Subtract \frac{47}{12} from both sides of the equation.

$$x=1 x=-\frac{53}{6}$$