Solve 2(3x-4)=7(11-2x)^2

$$2 ( 3 x - 4 ) = 7 ( 11 - 2 x ) ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{709} + 157}{28} \approx 6.558109068$$

$$x = \frac{157 - \sqrt{709}}{28} \approx 4.656176646$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2 by 3x-4.

$$6x-8=7\left(11-2x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(11-2x\right)^{2}.

$$6x-8=7\left(121-44x+4x^{2}\right)$$

Use the distributive property to multiply 7 by 121-44x+4x^{2}.

$$6x-8=847-308x+28x^{2}$$

Subtract 847 from both sides.

$$6x-8-847=-308x+28x^{2}$$

Subtract 847 from -8 to get -855.

$$6x-855=-308x+28x^{2}$$

Add 308x to both sides.

$$6x-855+308x=28x^{2}$$

Combine 6x and 308x to get 314x.

$$314x-855=28x^{2}$$

Subtract 28x^{2} from both sides.

$$314x-855-28x^{2}=0$$

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.

$$-28x^{2}+314x-855=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -28 for a, 314 for b, and -855 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-314±\sqrt{314^{2}-4\left(-28\right)\left(-855\right)}}{2\left(-28\right)}$$

Square 314.

$$x=\frac{-314±\sqrt{98596-4\left(-28\right)\left(-855\right)}}{2\left(-28\right)}$$

Multiply -4 times -28.

$$x=\frac{-314±\sqrt{98596+112\left(-855\right)}}{2\left(-28\right)}$$

Multiply 112 times -855.

$$x=\frac{-314±\sqrt{98596-95760}}{2\left(-28\right)}$$

Add 98596 to -95760.

$$x=\frac{-314±\sqrt{2836}}{2\left(-28\right)}$$

Take the square root of 2836.

$$x=\frac{-314±2\sqrt{709}}{2\left(-28\right)}$$

Multiply 2 times -28.

$$x=\frac{-314±2\sqrt{709}}{-56}$$

Now solve the equation x=\frac{-314±2\sqrt{709}}{-56} when ± is plus. Add -314 to 2\sqrt{709}.

$$x=\frac{2\sqrt{709}-314}{-56}$$

Divide -314+2\sqrt{709} by -56.

$$x=\frac{157-\sqrt{709}}{28}$$

Now solve the equation x=\frac{-314±2\sqrt{709}}{-56} when ± is minus. Subtract 2\sqrt{709} from -314.

$$x=\frac{-2\sqrt{709}-314}{-56}$$

Divide -314-2\sqrt{709} by -56.

$$x=\frac{\sqrt{709}+157}{28}$$

The equation is now solved.

$$x=\frac{157-\sqrt{709}}{28} x=\frac{\sqrt{709}+157}{28}$$

Steps for Completing the Square

Use the distributive property to multiply 2 by 3x-4.

$$6x-8=7\left(11-2x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(11-2x\right)^{2}.

$$6x-8=7\left(121-44x+4x^{2}\right)$$

Use the distributive property to multiply 7 by 121-44x+4x^{2}.

$$6x-8=847-308x+28x^{2}$$

Add 308x to both sides.

$$6x-8+308x=847+28x^{2}$$

Combine 6x and 308x to get 314x.

$$314x-8=847+28x^{2}$$

Subtract 28x^{2} from both sides.

$$314x-8-28x^{2}=847$$

Add 8 to both sides.

$$314x-28x^{2}=847+8$$

Add 847 and 8 to get 855.

$$314x-28x^{2}=855$$

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.

$$-28x^{2}+314x=855$$

Divide both sides by -28.

$$\frac{-28x^{2}+314x}{-28}=\frac{855}{-28}$$

Dividing by -28 undoes the multiplication by -28.

$$x^{2}+\frac{314}{-28}x=\frac{855}{-28}$$

Reduce the fraction \frac{314}{-28} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{157}{14}x=\frac{855}{-28}$$

Divide 855 by -28.

$$x^{2}-\frac{157}{14}x=-\frac{855}{28}$$

Divide -\frac{157}{14}, the coefficient of the x term, by 2 to get -\frac{157}{28}. Then add the square of -\frac{157}{28} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{157}{14}x+\left(-\frac{157}{28}\right)^{2}=-\frac{855}{28}+\left(-\frac{157}{28}\right)^{2}$$

Square -\frac{157}{28} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{157}{14}x+\frac{24649}{784}=-\frac{855}{28}+\frac{24649}{784}$$

Add -\frac{855}{28} to \frac{24649}{784} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{157}{14}x+\frac{24649}{784}=\frac{709}{784}$$

Factor x^{2}-\frac{157}{14}x+\frac{24649}{784}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{157}{28}\right)^{2}=\frac{709}{784}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{157}{28}\right)^{2}}=\sqrt{\frac{709}{784}}$$

Simplify.

$$x-\frac{157}{28}=\frac{\sqrt{709}}{28} x-\frac{157}{28}=-\frac{\sqrt{709}}{28}$$

Add \frac{157}{28} to both sides of the equation.

$$x=\frac{\sqrt{709}+157}{28} x=\frac{157-\sqrt{709}}{28}$$