Solve (x+3)*(x+4)+(x-2)*(x-1)=30

$$(x+3) \times (x+4)+(x-2) \times (x-1)=30$$

Solve for x

$$x=-4$$

$$x=2$$

Steps Using the Quadratic Formula

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

$$x^{2}+7x+12+\left(x-2\right)\left(x-1\right)=30$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$x^{2}+7x+12+x^{2}-3x+2=30$$

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

$$2x^{2}+7x+12-3x+2=30$$

Combine 7x and -3x to get 4x.

$$2x^{2}+4x+12+2=30$$

Add 12 and 2 to get 14.

$$2x^{2}+4x+14=30$$

Subtract 30 from both sides.

$$2x^{2}+4x+14-30=0$$

Subtract 30 from 14 to get -16.

$$2x^{2}+4x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 2\left(-16\right)}}{2\times 2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\left(-16\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\left(-16\right)}}{2\times 2}$$

Multiply -8 times -16.

$$x=\frac{-4±\sqrt{16+128}}{2\times 2}$$

Add 16 to 128.

$$x=\frac{-4±\sqrt{144}}{2\times 2}$$

Take the square root of 144.

$$x=\frac{-4±12}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±12}{4}$$

Now solve the equation x=\frac{-4±12}{4} when ± is plus. Add -4 to 12.

$$x=\frac{8}{4}$$

Divide 8 by 4.

$$x=2$$

Now solve the equation x=\frac{-4±12}{4} when ± is minus. Subtract 12 from -4.

$$x=-\frac{16}{4}$$

Divide -16 by 4.

$$x=-4$$

The equation is now solved.

$$x=2 x=-4$$

Steps for Completing the Square

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

$$x^{2}+7x+12+\left(x-2\right)\left(x-1\right)=30$$

Use the distributive property to multiply x-2 by x-1 and combine like terms.

$$x^{2}+7x+12+x^{2}-3x+2=30$$

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

$$2x^{2}+7x+12-3x+2=30$$

Combine 7x and -3x to get 4x.

$$2x^{2}+4x+12+2=30$$

Add 12 and 2 to get 14.

$$2x^{2}+4x+14=30$$

Subtract 14 from both sides.

$$2x^{2}+4x=30-14$$

Subtract 14 from 30 to get 16.

$$2x^{2}+4x=16$$

Divide both sides by 2.

$$\frac{2x^{2}+4x}{2}=\frac{16}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{4}{2}x=\frac{16}{2}$$

Divide 4 by 2.

$$x^{2}+2x=\frac{16}{2}$$

Divide 16 by 2.

$$x^{2}+2x=8$$

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2x+1^{2}=8+1^{2}$$

Square 1.

$$x^{2}+2x+1=8+1$$

Add 8 to 1.

$$x^{2}+2x+1=9$$

Factor x^{2}+2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{9}$$

Simplify.

$$x+1=3 x+1=-3$$

Subtract 1 from both sides of the equation.

$$x=2 x=-4$$