Solve 3x+(x+3)(x-4)=10

$$3x+(x+3)(x-4)=10$$

Solve for x (complex solution)

$$x=\sqrt{23}-1\approx 3.795831523$$

$$x=-\left(\sqrt{23}+1\right)\approx -5.795831523$$

Steps Using the Quadratic Formula

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

$$3x+x^{2}-x-12=10$$

Combine 3x and -x to get 2x.

$$2x+x^{2}-12=10$$

Subtract 10 from both sides.

$$2x+x^{2}-12-10=0$$

Subtract 10 from -12 to get -22.

$$2x+x^{2}-22=0$$

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.

$$x^{2}+2x-22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-22\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-22\right)}}{2}$$

Multiply -4 times -22.

$$x=\frac{-2±\sqrt{4+88}}{2}$$

Add 4 to 88.

$$x=\frac{-2±\sqrt{92}}{2}$$

Take the square root of 92.

$$x=\frac{-2±2\sqrt{23}}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{23}}{2} when ± is plus. Add -2 to 2\sqrt{23}.

$$x=\frac{2\sqrt{23}-2}{2}$$

Divide -2+2\sqrt{23} by 2.

$$x=\sqrt{23}-1$$

Now solve the equation x=\frac{-2±2\sqrt{23}}{2} when ± is minus. Subtract 2\sqrt{23} from -2.

$$x=\frac{-2\sqrt{23}-2}{2}$$

Divide -2-2\sqrt{23} by 2.

$$x=-\sqrt{23}-1$$

The equation is now solved.

$$x=\sqrt{23}-1 x=-\sqrt{23}-1$$

Steps for Completing the Square

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

$$3x+x^{2}-x-12=10$$

Combine 3x and -x to get 2x.

$$2x+x^{2}-12=10$$

Add 12 to both sides.

$$2x+x^{2}=10+12$$

Add 10 and 12 to get 22.

$$2x+x^{2}=22$$

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.

$$x^{2}+2x=22$$

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.

$$x^{2}+2x+1^{2}=22+1^{2}$$

Square 1.

$$x^{2}+2x+1=22+1$$

Add 22 to 1.

$$x^{2}+2x+1=23$$

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}.

$$\left(x+1\right)^{2}=23$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{23}$$

Simplify.

$$x+1=\sqrt{23} x+1=-\sqrt{23}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{23}-1 x=-\sqrt{23}-1$$

Solve for x

$$x=\sqrt{23}-1\approx 3.795831523$$

$$x=-\sqrt{23}-1\approx -5.795831523$$

Steps Using the Quadratic Formula

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

$$3x+x^{2}-x-12=10$$

Combine 3x and -x to get 2x.

$$2x+x^{2}-12=10$$

Subtract 10 from both sides.

$$2x+x^{2}-12-10=0$$

Subtract 10 from -12 to get -22.

$$2x+x^{2}-22=0$$

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.

$$x^{2}+2x-22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-22\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-22\right)}}{2}$$

Multiply -4 times -22.

$$x=\frac{-2±\sqrt{4+88}}{2}$$

Add 4 to 88.

$$x=\frac{-2±\sqrt{92}}{2}$$

Take the square root of 92.

$$x=\frac{-2±2\sqrt{23}}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{23}}{2} when ± is plus. Add -2 to 2\sqrt{23}.

$$x=\frac{2\sqrt{23}-2}{2}$$

Divide -2+2\sqrt{23} by 2.

$$x=\sqrt{23}-1$$

Now solve the equation x=\frac{-2±2\sqrt{23}}{2} when ± is minus. Subtract 2\sqrt{23} from -2.

$$x=\frac{-2\sqrt{23}-2}{2}$$

Divide -2-2\sqrt{23} by 2.

$$x=-\sqrt{23}-1$$

The equation is now solved.

$$x=\sqrt{23}-1 x=-\sqrt{23}-1$$

Steps for Completing the Square

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

$$3x+x^{2}-x-12=10$$

Combine 3x and -x to get 2x.

$$2x+x^{2}-12=10$$

Add 12 to both sides.

$$2x+x^{2}=10+12$$

Add 10 and 12 to get 22.

$$2x+x^{2}=22$$

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.

$$x^{2}+2x=22$$

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.

$$x^{2}+2x+1^{2}=22+1^{2}$$

Square 1.

$$x^{2}+2x+1=22+1$$

Add 22 to 1.

$$x^{2}+2x+1=23$$

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}.

$$\left(x+1\right)^{2}=23$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{23}$$

Simplify.

$$x+1=\sqrt{23} x+1=-\sqrt{23}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{23}-1 x=-\sqrt{23}-1$$