Solve left(3x-4right)^2-3x^2=2left(8+13xright)

$${ \left(3x-4 \right) }^{ 2 } -3 { x }^{ 2 } = 2 \left( 8+13x \right)$$

Solve for x

$$x = \frac{25}{3} = 8\frac{1}{3} \approx 8.333333333$$

$$x=0$$

Steps Using Factoring

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-4\right)^{2}.

$$9x^{2}-24x+16-3x^{2}=2\left(8+13x\right)$$

Combine 9x^{2} and -3x^{2} to get 6x^{2}.

$$6x^{2}-24x+16=2\left(8+13x\right)$$

Use the distributive property to multiply 2 by 8+13x.

$$6x^{2}-24x+16=16+26x$$

Subtract 16 from both sides.

$$6x^{2}-24x+16-16=26x$$

Subtract 16 from 16 to get 0.

$$6x^{2}-24x=26x$$

Subtract 26x from both sides.

$$6x^{2}-24x-26x=0$$

Combine -24x and -26x to get -50x.

$$6x^{2}-50x=0$$

Factor out x.

$$x\left(6x-50\right)=0$$

To find equation solutions, solve x=0 and 6x-50=0.

$$x=0 x=\frac{25}{3}$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-4\right)^{2}.

$$9x^{2}-24x+16-3x^{2}=2\left(8+13x\right)$$

Combine 9x^{2} and -3x^{2} to get 6x^{2}.

$$6x^{2}-24x+16=2\left(8+13x\right)$$

Use the distributive property to multiply 2 by 8+13x.

$$6x^{2}-24x+16=16+26x$$

Subtract 16 from both sides.

$$6x^{2}-24x+16-16=26x$$

Subtract 16 from 16 to get 0.

$$6x^{2}-24x=26x$$

Subtract 26x from both sides.

$$6x^{2}-24x-26x=0$$

Combine -24x and -26x to get -50x.

$$6x^{2}-50x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -50 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}}}{2\times 6}$$

Take the square root of \left(-50\right)^{2}.

$$x=\frac{-\left(-50\right)±50}{2\times 6}$$

The opposite of -50 is 50.

$$x=\frac{50±50}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{50±50}{12}$$

Now solve the equation x=\frac{50±50}{12} when ± is plus. Add 50 to 50.

$$x=\frac{100}{12}$$

Reduce the fraction \frac{100}{12} to lowest terms by extracting and canceling out 4.

$$x=\frac{25}{3}$$

Now solve the equation x=\frac{50±50}{12} when ± is minus. Subtract 50 from 50.

$$x=\frac{0}{12}$$

Divide 0 by 12.

$$x=0$$

The equation is now solved.

$$x=\frac{25}{3} x=0$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-4\right)^{2}.

$$9x^{2}-24x+16-3x^{2}=2\left(8+13x\right)$$

Combine 9x^{2} and -3x^{2} to get 6x^{2}.

$$6x^{2}-24x+16=2\left(8+13x\right)$$

Use the distributive property to multiply 2 by 8+13x.

$$6x^{2}-24x+16=16+26x$$

Subtract 26x from both sides.

$$6x^{2}-24x+16-26x=16$$

Combine -24x and -26x to get -50x.

$$6x^{2}-50x+16=16$$

Subtract 16 from both sides.

$$6x^{2}-50x=16-16$$

Subtract 16 from 16 to get 0.

$$6x^{2}-50x=0$$

Divide both sides by 6.

$$\frac{6x^{2}-50x}{6}=\frac{0}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{50}{6}\right)x=\frac{0}{6}$$

Reduce the fraction \frac{-50}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{25}{3}x=\frac{0}{6}$$

Divide 0 by 6.

$$x^{2}-\frac{25}{3}x=0$$

Divide -\frac{25}{3}, the coefficient of the x term, by 2 to get -\frac{25}{6}. Then add the square of -\frac{25}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{3}x+\left(-\frac{25}{6}\right)^{2}=\left(-\frac{25}{6}\right)^{2}$$

Square -\frac{25}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{3}x+\frac{625}{36}=\frac{625}{36}$$

Factor x^{2}-\frac{25}{3}x+\frac{625}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{6}\right)^{2}=\frac{625}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{6}\right)^{2}}=\sqrt{\frac{625}{36}}$$

Simplify.

$$x-\frac{25}{6}=\frac{25}{6} x-\frac{25}{6}=-\frac{25}{6}$$

Add \frac{25}{6} to both sides of the equation.

$$x=\frac{25}{3} x=0$$