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

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

Solve for x

$$x=-11$$

$$x=2$$

Steps Using the Quadratic Formula

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

$$9x-6=\left(4-x\right)\left(4+x\right)$$

Consider \left(4-x\right)\left(4+x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 4.

$$9x-6=16-x^{2}$$

Subtract 16 from both sides.

$$9x-6-16=-x^{2}$$

Subtract 16 from -6 to get -22.

$$9x-22=-x^{2}$$

Add x^{2} to both sides.

$$9x-22+x^{2}=0$$

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.

$$x^{2}+9x-22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 9 for b, and -22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-22\right)}}{2}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\left(-22\right)}}{2}$$

Multiply -4 times -22.

$$x=\frac{-9±\sqrt{81+88}}{2}$$

Add 81 to 88.

$$x=\frac{-9±\sqrt{169}}{2}$$

Take the square root of 169.

$$x=\frac{-9±13}{2}$$

Now solve the equation x=\frac{-9±13}{2} when ± is plus. Add -9 to 13.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{-9±13}{2} when ± is minus. Subtract 13 from -9.

$$x=-\frac{22}{2}$$

Divide -22 by 2.

$$x=-11$$

The equation is now solved.

$$x=2 x=-11$$

Steps for Completing the Square

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

$$9x-6=\left(4-x\right)\left(4+x\right)$$

Consider \left(4-x\right)\left(4+x\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 4.

$$9x-6=16-x^{2}$$

Add x^{2} to both sides.

$$9x-6+x^{2}=16$$

Add 6 to both sides.

$$9x+x^{2}=16+6$$

Add 16 and 6 to get 22.

$$9x+x^{2}=22$$

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.

$$x^{2}+9x=22$$

Divide 9, the coefficient of the x term, by 2 to get \frac{9}{2}. Then add the square of \frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+9x+\left(\frac{9}{2}\right)^{2}=22+\left(\frac{9}{2}\right)^{2}$$

Square \frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+9x+\frac{81}{4}=22+\frac{81}{4}$$

Add 22 to \frac{81}{4}.

$$x^{2}+9x+\frac{81}{4}=\frac{169}{4}$$

Factor x^{2}+9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{9}{2}\right)^{2}=\frac{169}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Simplify.

$$x+\frac{9}{2}=\frac{13}{2} x+\frac{9}{2}=-\frac{13}{2}$$

Subtract \frac{9}{2} from both sides of the equation.

$$x=2 x=-11$$