Solve (x-4)(x+7)-(2x+3)(x+7)=0

$$(x-4)(x+7)-(2x+3)(x+7)=0$$

Solve for x

$$x=-7$$

Steps Using Factoring By Grouping

Use the distributive property to multiply x-4 by x+7 and combine like terms.

$$x^{2}+3x-28-\left(2x+3\right)\left(x+7\right)=0$$

Use the distributive property to multiply 2x+3 by x+7 and combine like terms.

$$x^{2}+3x-28-\left(2x^{2}+17x+21\right)=0$$

To find the opposite of 2x^{2}+17x+21, find the opposite of each term.

$$x^{2}+3x-28-2x^{2}-17x-21=0$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}+3x-28-17x-21=0$$

Combine 3x and -17x to get -14x.

$$-x^{2}-14x-28-21=0$$

Subtract 21 from -28 to get -49.

$$-x^{2}-14x-49=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-49. To find a and b, set up a system to be solved.

$$a+b=-14 ab=-\left(-49\right)=49$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 49.

$$-1,-49 -7,-7$$

Calculate the sum for each pair.

$$-1-49=-50 -7-7=-14$$

The solution is the pair that gives sum -14.

$$a=-7 b=-7$$

Rewrite -x^{2}-14x-49 as \left(-x^{2}-7x\right)+\left(-7x-49\right).

$$\left(-x^{2}-7x\right)+\left(-7x-49\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(-x-7\right)+7\left(-x-7\right)$$

Factor out common term -x-7 by using distributive property.

$$\left(-x-7\right)\left(x+7\right)$$

To find equation solutions, solve -x-7=0 and x+7=0.

$$x=-7 x=-7$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-4 by x+7 and combine like terms.

$$x^{2}+3x-28-\left(2x+3\right)\left(x+7\right)=0$$

Use the distributive property to multiply 2x+3 by x+7 and combine like terms.

$$x^{2}+3x-28-\left(2x^{2}+17x+21\right)=0$$

To find the opposite of 2x^{2}+17x+21, find the opposite of each term.

$$x^{2}+3x-28-2x^{2}-17x-21=0$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}+3x-28-17x-21=0$$

Combine 3x and -17x to get -14x.

$$-x^{2}-14x-28-21=0$$

Subtract 21 from -28 to get -49.

$$-x^{2}-14x-49=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -14 for b, and -49 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-1\right)\left(-49\right)}}{2\left(-1\right)}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-1\right)\left(-49\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-14\right)±\sqrt{196+4\left(-49\right)}}{2\left(-1\right)}$$

Multiply 4 times -49.

$$x=\frac{-\left(-14\right)±\sqrt{196-196}}{2\left(-1\right)}$$

Add 196 to -196.

$$x=\frac{-\left(-14\right)±\sqrt{0}}{2\left(-1\right)}$$

Take the square root of 0.

$$x=-\frac{-14}{2\left(-1\right)}$$

The opposite of -14 is 14.

$$x=\frac{14}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{14}{-2}$$

Divide 14 by -2.

$$x=-7$$

Steps for Completing the Square

Use the distributive property to multiply x-4 by x+7 and combine like terms.

$$x^{2}+3x-28-\left(2x+3\right)\left(x+7\right)=0$$

Use the distributive property to multiply 2x+3 by x+7 and combine like terms.

$$x^{2}+3x-28-\left(2x^{2}+17x+21\right)=0$$

To find the opposite of 2x^{2}+17x+21, find the opposite of each term.

$$x^{2}+3x-28-2x^{2}-17x-21=0$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}+3x-28-17x-21=0$$

Combine 3x and -17x to get -14x.

$$-x^{2}-14x-28-21=0$$

Subtract 21 from -28 to get -49.

$$-x^{2}-14x-49=0$$

Add 49 to both sides. Anything plus zero gives itself.

$$-x^{2}-14x=49$$

Divide both sides by -1.

$$\frac{-x^{2}-14x}{-1}=\frac{49}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{14}{-1}\right)x=\frac{49}{-1}$$

Divide -14 by -1.

$$x^{2}+14x=\frac{49}{-1}$$

Divide 49 by -1.

$$x^{2}+14x=-49$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+14x+7^{2}=-49+7^{2}$$

Square 7.

$$x^{2}+14x+49=-49+49$$

Add -49 to 49.

$$x^{2}+14x+49=0$$

Factor x^{2}+14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+7\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{0}$$

Simplify.

$$x+7=0 x+7=0$$

Subtract 7 from both sides of the equation.

$$x=-7 x=-7$$

The equation is now solved. Solutions are the same.

$$x=-7$$