Solve left(x-7right)^2+18=left(4-xright)left(x+4right)

$${ \left(x-7 \right) }^{ 2 } +18 = \left( 4-x \right) \left( x+4 \right)$$

Solve for x (complex solution)

$$x=\frac{7+\sqrt{53}i}{2}\approx 3.5+3.640054945i$$

$$x=\frac{-\sqrt{53}i+7}{2}\approx 3.5-3.640054945i$$

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}.

$$x^{2}-14x+49+18=\left(4-x\right)\left(x+4\right)$$

Add 49 and 18 to get 67.

$$x^{2}-14x+67=\left(4-x\right)\left(x+4\right)$$

Consider \left(4-x\right)\left(x+4\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.

$$x^{2}-14x+67=16-x^{2}$$

Subtract 16 from both sides.

$$x^{2}-14x+67-16=-x^{2}$$

Subtract 16 from 67 to get 51.

$$x^{2}-14x+51=-x^{2}$$

Add x^{2} to both sides.

$$x^{2}-14x+51+x^{2}=0$$

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

$$2x^{2}-14x+51=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -14 for b, and 51 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 2\times 51}}{2\times 2}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 2\times 51}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-14\right)±\sqrt{196-8\times 51}}{2\times 2}$$

Multiply -8 times 51.

$$x=\frac{-\left(-14\right)±\sqrt{196-408}}{2\times 2}$$

Add 196 to -408.

$$x=\frac{-\left(-14\right)±\sqrt{-212}}{2\times 2}$$

Take the square root of -212.

$$x=\frac{-\left(-14\right)±2\sqrt{53}i}{2\times 2}$$

The opposite of -14 is 14.

$$x=\frac{14±2\sqrt{53}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{14±2\sqrt{53}i}{4}$$

Now solve the equation x=\frac{14±2\sqrt{53}i}{4} when ± is plus. Add 14 to 2i\sqrt{53}.

$$x=\frac{14+2\sqrt{53}i}{4}$$

Divide 14+2i\sqrt{53} by 4.

$$x=\frac{7+\sqrt{53}i}{2}$$

Now solve the equation x=\frac{14±2\sqrt{53}i}{4} when ± is minus. Subtract 2i\sqrt{53} from 14.

$$x=\frac{-2\sqrt{53}i+14}{4}$$

Divide 14-2i\sqrt{53} by 4.

$$x=\frac{-\sqrt{53}i+7}{2}$$

The equation is now solved.

$$x=\frac{7+\sqrt{53}i}{2} x=\frac{-\sqrt{53}i+7}{2}$$

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}.

$$x^{2}-14x+49+18=\left(4-x\right)\left(x+4\right)$$

Add 49 and 18 to get 67.

$$x^{2}-14x+67=\left(4-x\right)\left(x+4\right)$$

Consider \left(4-x\right)\left(x+4\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.

$$x^{2}-14x+67=16-x^{2}$$

Add x^{2} to both sides.

$$x^{2}-14x+67+x^{2}=16$$

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

$$2x^{2}-14x+67=16$$

Subtract 67 from both sides.

$$2x^{2}-14x=16-67$$

Subtract 67 from 16 to get -51.

$$2x^{2}-14x=-51$$

Divide both sides by 2.

$$\frac{2x^{2}-14x}{2}=-\frac{51}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{14}{2}\right)x=-\frac{51}{2}$$

Divide -14 by 2.

$$x^{2}-7x=-\frac{51}{2}$$

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

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-\frac{51}{2}+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-\frac{51}{2}+\frac{49}{4}$$

Add -\frac{51}{2} to \frac{49}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-7x+\frac{49}{4}=-\frac{53}{4}$$

Factor x^{2}-7x+\frac{49}{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}.

$$\left(x-\frac{7}{2}\right)^{2}=-\frac{53}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{-\frac{53}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{\sqrt{53}i}{2} x-\frac{7}{2}=-\frac{\sqrt{53}i}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=\frac{7+\sqrt{53}i}{2} x=\frac{-\sqrt{53}i+7}{2}$$