Solve (7x+21)(-5x+8)=4(2x-7)(4x+12)

$$( 7 x + 21 ) ( - 5 x + 8 ) = 4 ( 2 x - 7 ) ( 4 x + 12 )$$

Solve for x

$$x = \frac{168}{67} = 2\frac{34}{67} \approx 2.507462687$$

$$x=-3$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 7x+21 by -5x+8 and combine like terms.

$$-35x^{2}-49x+168=4\left(2x-7\right)\left(4x+12\right)$$

Use the distributive property to multiply 4 by 2x-7.

$$-35x^{2}-49x+168=\left(8x-28\right)\left(4x+12\right)$$

Use the distributive property to multiply 8x-28 by 4x+12 and combine like terms.

$$-35x^{2}-49x+168=32x^{2}-16x-336$$

Subtract 32x^{2} from both sides.

$$-35x^{2}-49x+168-32x^{2}=-16x-336$$

Combine -35x^{2} and -32x^{2} to get -67x^{2}.

$$-67x^{2}-49x+168=-16x-336$$

Add 16x to both sides.

$$-67x^{2}-49x+168+16x=-336$$

Combine -49x and 16x to get -33x.

$$-67x^{2}-33x+168=-336$$

Add 336 to both sides.

$$-67x^{2}-33x+168+336=0$$

Add 168 and 336 to get 504.

$$-67x^{2}-33x+504=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -67 for a, -33 for b, and 504 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\left(-67\right)\times 504}}{2\left(-67\right)}$$

Square -33.

$$x=\frac{-\left(-33\right)±\sqrt{1089-4\left(-67\right)\times 504}}{2\left(-67\right)}$$

Multiply -4 times -67.

$$x=\frac{-\left(-33\right)±\sqrt{1089+268\times 504}}{2\left(-67\right)}$$

Multiply 268 times 504.

$$x=\frac{-\left(-33\right)±\sqrt{1089+135072}}{2\left(-67\right)}$$

Add 1089 to 135072.

$$x=\frac{-\left(-33\right)±\sqrt{136161}}{2\left(-67\right)}$$

Take the square root of 136161.

$$x=\frac{-\left(-33\right)±369}{2\left(-67\right)}$$

The opposite of -33 is 33.

$$x=\frac{33±369}{2\left(-67\right)}$$

Multiply 2 times -67.

$$x=\frac{33±369}{-134}$$

Now solve the equation x=\frac{33±369}{-134} when ± is plus. Add 33 to 369.

$$x=\frac{402}{-134}$$

Divide 402 by -134.

$$x=-3$$

Now solve the equation x=\frac{33±369}{-134} when ± is minus. Subtract 369 from 33.

$$x=-\frac{336}{-134}$$

Reduce the fraction \frac{-336}{-134} to lowest terms by extracting and canceling out 2.

$$x=\frac{168}{67}$$

The equation is now solved.

$$x=-3 x=\frac{168}{67}$$

Steps for Completing the Square

Use the distributive property to multiply 7x+21 by -5x+8 and combine like terms.

$$-35x^{2}-49x+168=4\left(2x-7\right)\left(4x+12\right)$$

Use the distributive property to multiply 4 by 2x-7.

$$-35x^{2}-49x+168=\left(8x-28\right)\left(4x+12\right)$$

Use the distributive property to multiply 8x-28 by 4x+12 and combine like terms.

$$-35x^{2}-49x+168=32x^{2}-16x-336$$

Subtract 32x^{2} from both sides.

$$-35x^{2}-49x+168-32x^{2}=-16x-336$$

Combine -35x^{2} and -32x^{2} to get -67x^{2}.

$$-67x^{2}-49x+168=-16x-336$$

Add 16x to both sides.

$$-67x^{2}-49x+168+16x=-336$$

Combine -49x and 16x to get -33x.

$$-67x^{2}-33x+168=-336$$

Subtract 168 from both sides.

$$-67x^{2}-33x=-336-168$$

Subtract 168 from -336 to get -504.

$$-67x^{2}-33x=-504$$

Divide both sides by -67.

$$\frac{-67x^{2}-33x}{-67}=-\frac{504}{-67}$$

Dividing by -67 undoes the multiplication by -67.

$$x^{2}+\left(-\frac{33}{-67}\right)x=-\frac{504}{-67}$$

Divide -33 by -67.

$$x^{2}+\frac{33}{67}x=-\frac{504}{-67}$$

Divide -504 by -67.

$$x^{2}+\frac{33}{67}x=\frac{504}{67}$$

Divide \frac{33}{67}, the coefficient of the x term, by 2 to get \frac{33}{134}. Then add the square of \frac{33}{134} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{33}{67}x+\left(\frac{33}{134}\right)^{2}=\frac{504}{67}+\left(\frac{33}{134}\right)^{2}$$

Square \frac{33}{134} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{33}{67}x+\frac{1089}{17956}=\frac{504}{67}+\frac{1089}{17956}$$

Add \frac{504}{67} to \frac{1089}{17956} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{33}{67}x+\frac{1089}{17956}=\frac{136161}{17956}$$

Factor x^{2}+\frac{33}{67}x+\frac{1089}{17956}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{33}{134}\right)^{2}=\frac{136161}{17956}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{33}{134}\right)^{2}}=\sqrt{\frac{136161}{17956}}$$

Simplify.

$$x+\frac{33}{134}=\frac{369}{134} x+\frac{33}{134}=-\frac{369}{134}$$

Subtract \frac{33}{134} from both sides of the equation.

$$x=\frac{168}{67} x=-3$$