Solve left(3x-8right)^2+left(-8+3xright)left(4x+2right)=48

$${ \left(3x-8 \right) }^{ 2 } + \left( -8+3x \right) \left( 4x+2 \right) =48$$

Solve for x

$$x = \frac{74}{21} = 3\frac{11}{21} \approx 3.523809524$$

$$x=0$$

Steps Using Factoring

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-8\right)^{2}.

$$9x^{2}-48x+64+\left(-8+3x\right)\left(4x+2\right)=48$$

Use the distributive property to multiply -8+3x by 4x+2 and combine like terms.

$$9x^{2}-48x+64-26x-16+12x^{2}=48$$

Combine -48x and -26x to get -74x.

$$9x^{2}-74x+64-16+12x^{2}=48$$

Subtract 16 from 64 to get 48.

$$9x^{2}-74x+48+12x^{2}=48$$

Combine 9x^{2} and 12x^{2} to get 21x^{2}.

$$21x^{2}-74x+48=48$$

Subtract 48 from both sides.

$$21x^{2}-74x+48-48=0$$

Subtract 48 from 48 to get 0.

$$21x^{2}-74x=0$$

Factor out x.

$$x\left(21x-74\right)=0$$

To find equation solutions, solve x=0 and 21x-74=0.

$$x=0 x=\frac{74}{21}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-8\right)^{2}.

$$9x^{2}-48x+64+\left(-8+3x\right)\left(4x+2\right)=48$$

Use the distributive property to multiply -8+3x by 4x+2 and combine like terms.

$$9x^{2}-48x+64-26x-16+12x^{2}=48$$

Combine -48x and -26x to get -74x.

$$9x^{2}-74x+64-16+12x^{2}=48$$

Subtract 16 from 64 to get 48.

$$9x^{2}-74x+48+12x^{2}=48$$

Combine 9x^{2} and 12x^{2} to get 21x^{2}.

$$21x^{2}-74x+48=48$$

Subtract 48 from both sides.

$$21x^{2}-74x+48-48=0$$

Subtract 48 from 48 to get 0.

$$21x^{2}-74x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 21 for a, -74 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-74\right)±\sqrt{\left(-74\right)^{2}}}{2\times 21}$$

Take the square root of \left(-74\right)^{2}.

$$x=\frac{-\left(-74\right)±74}{2\times 21}$$

The opposite of -74 is 74.

$$x=\frac{74±74}{2\times 21}$$

Multiply 2 times 21.

$$x=\frac{74±74}{42}$$

Now solve the equation x=\frac{74±74}{42} when ± is plus. Add 74 to 74.

$$x=\frac{148}{42}$$

Reduce the fraction \frac{148}{42} to lowest terms by extracting and canceling out 2.

$$x=\frac{74}{21}$$

Now solve the equation x=\frac{74±74}{42} when ± is minus. Subtract 74 from 74.

$$x=\frac{0}{42}$$

Divide 0 by 42.

$$x=0$$

The equation is now solved.

$$x=\frac{74}{21} x=0$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-8\right)^{2}.

$$9x^{2}-48x+64+\left(-8+3x\right)\left(4x+2\right)=48$$

Use the distributive property to multiply -8+3x by 4x+2 and combine like terms.

$$9x^{2}-48x+64-26x-16+12x^{2}=48$$

Combine -48x and -26x to get -74x.

$$9x^{2}-74x+64-16+12x^{2}=48$$

Subtract 16 from 64 to get 48.

$$9x^{2}-74x+48+12x^{2}=48$$

Combine 9x^{2} and 12x^{2} to get 21x^{2}.

$$21x^{2}-74x+48=48$$

Subtract 48 from both sides.

$$21x^{2}-74x=48-48$$

Subtract 48 from 48 to get 0.

$$21x^{2}-74x=0$$

Divide both sides by 21.

$$\frac{21x^{2}-74x}{21}=\frac{0}{21}$$

Dividing by 21 undoes the multiplication by 21.

$$x^{2}-\frac{74}{21}x=\frac{0}{21}$$

Divide 0 by 21.

$$x^{2}-\frac{74}{21}x=0$$

Divide -\frac{74}{21}, the coefficient of the x term, by 2 to get -\frac{37}{21}. Then add the square of -\frac{37}{21} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{74}{21}x+\left(-\frac{37}{21}\right)^{2}=\left(-\frac{37}{21}\right)^{2}$$

Square -\frac{37}{21} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{74}{21}x+\frac{1369}{441}=\frac{1369}{441}$$

Factor x^{2}-\frac{74}{21}x+\frac{1369}{441}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{37}{21}\right)^{2}=\frac{1369}{441}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{37}{21}\right)^{2}}=\sqrt{\frac{1369}{441}}$$

Simplify.

$$x-\frac{37}{21}=\frac{37}{21} x-\frac{37}{21}=-\frac{37}{21}$$

Add \frac{37}{21} to both sides of the equation.

$$x=\frac{74}{21} x=0$$