Solve (x)=3(x-5)(x+3)

$$( x ) = 3 ( x - 5 ) ( x + 3 )$$

Solve for x

$$x = \frac{\sqrt{589} + 7}{6} \approx 5.2115537$$

$$x=\frac{7-\sqrt{589}}{6}\approx -2.878220367$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3 by x-5.

$$x=\left(3x-15\right)\left(x+3\right)$$

Use the distributive property to multiply 3x-15 by x+3 and combine like terms.

$$x=3x^{2}-6x-45$$

Subtract 3x^{2} from both sides.

$$x-3x^{2}=-6x-45$$

Add 6x to both sides.

$$x-3x^{2}+6x=-45$$

Combine x and 6x to get 7x.

$$7x-3x^{2}=-45$$

Add 45 to both sides.

$$7x-3x^{2}+45=0$$

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.

$$-3x^{2}+7x+45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 7 for b, and 45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-3\right)\times 45}}{2\left(-3\right)}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-3\right)\times 45}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-7±\sqrt{49+12\times 45}}{2\left(-3\right)}$$

Multiply 12 times 45.

$$x=\frac{-7±\sqrt{49+540}}{2\left(-3\right)}$$

Add 49 to 540.

$$x=\frac{-7±\sqrt{589}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-7±\sqrt{589}}{-6}$$

Now solve the equation x=\frac{-7±\sqrt{589}}{-6} when ± is plus. Add -7 to \sqrt{589}.

$$x=\frac{\sqrt{589}-7}{-6}$$

Divide -7+\sqrt{589} by -6.

$$x=\frac{7-\sqrt{589}}{6}$$

Now solve the equation x=\frac{-7±\sqrt{589}}{-6} when ± is minus. Subtract \sqrt{589} from -7.

$$x=\frac{-\sqrt{589}-7}{-6}$$

Divide -7-\sqrt{589} by -6.

$$x=\frac{\sqrt{589}+7}{6}$$

The equation is now solved.

$$x=\frac{7-\sqrt{589}}{6} x=\frac{\sqrt{589}+7}{6}$$

Steps for Completing the Square

Use the distributive property to multiply 3 by x-5.

$$x=\left(3x-15\right)\left(x+3\right)$$

Use the distributive property to multiply 3x-15 by x+3 and combine like terms.

$$x=3x^{2}-6x-45$$

Subtract 3x^{2} from both sides.

$$x-3x^{2}=-6x-45$$

Add 6x to both sides.

$$x-3x^{2}+6x=-45$$

Combine x and 6x to get 7x.

$$7x-3x^{2}=-45$$

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.

$$-3x^{2}+7x=-45$$

Divide both sides by -3.

$$\frac{-3x^{2}+7x}{-3}=-\frac{45}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{7}{-3}x=-\frac{45}{-3}$$

Divide 7 by -3.

$$x^{2}-\frac{7}{3}x=-\frac{45}{-3}$$

Divide -45 by -3.

$$x^{2}-\frac{7}{3}x=15$$

Divide -\frac{7}{3}, the coefficient of the x term, by 2 to get -\frac{7}{6}. Then add the square of -\frac{7}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=15+\left(-\frac{7}{6}\right)^{2}$$

Square -\frac{7}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=15+\frac{49}{36}$$

Add 15 to \frac{49}{36}.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{589}{36}$$

Factor x^{2}-\frac{7}{3}x+\frac{49}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{6}\right)^{2}=\frac{589}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{\frac{589}{36}}$$

Simplify.

$$x-\frac{7}{6}=\frac{\sqrt{589}}{6} x-\frac{7}{6}=-\frac{\sqrt{589}}{6}$$

Add \frac{7}{6} to both sides of the equation.

$$x=\frac{\sqrt{589}+7}{6} x=\frac{7-\sqrt{589}}{6}$$