Solve (x)=3(x+7)^2-27

$$( x ) = 3 ( x + 7 ) ^ { 2 } - 27$$

Solve for x

$$x=\frac{\sqrt{241}-41}{6}\approx -4.245970884$$

$$x=\frac{-\sqrt{241}-41}{6}\approx -9.420695783$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x=3\left(x^{2}+14x+49\right)-27$$

Use the distributive property to multiply 3 by x^{2}+14x+49.

$$x=3x^{2}+42x+147-27$$

Subtract 27 from 147 to get 120.

$$x=3x^{2}+42x+120$$

Subtract 3x^{2} from both sides.

$$x-3x^{2}=42x+120$$

Subtract 42x from both sides.

$$x-3x^{2}-42x=120$$

Combine x and -42x to get -41x.

$$-41x-3x^{2}=120$$

Subtract 120 from both sides.

$$-41x-3x^{2}-120=0$$

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.

$$-3x^{2}-41x-120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, -41 for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-41\right)±\sqrt{\left(-41\right)^{2}-4\left(-3\right)\left(-120\right)}}{2\left(-3\right)}$$

Square -41.

$$x=\frac{-\left(-41\right)±\sqrt{1681-4\left(-3\right)\left(-120\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-\left(-41\right)±\sqrt{1681+12\left(-120\right)}}{2\left(-3\right)}$$

Multiply 12 times -120.

$$x=\frac{-\left(-41\right)±\sqrt{1681-1440}}{2\left(-3\right)}$$

Add 1681 to -1440.

$$x=\frac{-\left(-41\right)±\sqrt{241}}{2\left(-3\right)}$$

The opposite of -41 is 41.

$$x=\frac{41±\sqrt{241}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{41±\sqrt{241}}{-6}$$

Now solve the equation x=\frac{41±\sqrt{241}}{-6} when ± is plus. Add 41 to \sqrt{241}.

$$x=\frac{\sqrt{241}+41}{-6}$$

Divide 41+\sqrt{241} by -6.

$$x=\frac{-\sqrt{241}-41}{6}$$

Now solve the equation x=\frac{41±\sqrt{241}}{-6} when ± is minus. Subtract \sqrt{241} from 41.

$$x=\frac{41-\sqrt{241}}{-6}$$

Divide 41-\sqrt{241} by -6.

$$x=\frac{\sqrt{241}-41}{6}$$

The equation is now solved.

$$x=\frac{-\sqrt{241}-41}{6} x=\frac{\sqrt{241}-41}{6}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x=3\left(x^{2}+14x+49\right)-27$$

Use the distributive property to multiply 3 by x^{2}+14x+49.

$$x=3x^{2}+42x+147-27$$

Subtract 27 from 147 to get 120.

$$x=3x^{2}+42x+120$$

Subtract 3x^{2} from both sides.

$$x-3x^{2}=42x+120$$

Subtract 42x from both sides.

$$x-3x^{2}-42x=120$$

Combine x and -42x to get -41x.

$$-41x-3x^{2}=120$$

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.

$$-3x^{2}-41x=120$$

Divide both sides by -3.

$$\frac{-3x^{2}-41x}{-3}=\frac{120}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\left(-\frac{41}{-3}\right)x=\frac{120}{-3}$$

Divide -41 by -3.

$$x^{2}+\frac{41}{3}x=\frac{120}{-3}$$

Divide 120 by -3.

$$x^{2}+\frac{41}{3}x=-40$$

Divide \frac{41}{3}, the coefficient of the x term, by 2 to get \frac{41}{6}. Then add the square of \frac{41}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{41}{3}x+\left(\frac{41}{6}\right)^{2}=-40+\left(\frac{41}{6}\right)^{2}$$

Square \frac{41}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{41}{3}x+\frac{1681}{36}=-40+\frac{1681}{36}$$

Add -40 to \frac{1681}{36}.

$$x^{2}+\frac{41}{3}x+\frac{1681}{36}=\frac{241}{36}$$

Factor x^{2}+\frac{41}{3}x+\frac{1681}{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}.

$$\left(x+\frac{41}{6}\right)^{2}=\frac{241}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{41}{6}\right)^{2}}=\sqrt{\frac{241}{36}}$$

Simplify.

$$x+\frac{41}{6}=\frac{\sqrt{241}}{6} x+\frac{41}{6}=-\frac{\sqrt{241}}{6}$$

Subtract \frac{41}{6} from both sides of the equation.

$$x=\frac{\sqrt{241}-41}{6} x=\frac{-\sqrt{241}-41}{6}$$