Solve x=(x^2-4)+8(x+4)-7

$$x=( { x }^{ 2 } -4)+8(x+4)-7$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{35}i-7}{2}\approx -3.5-2.958039892i$$

$$x=\frac{-7+\sqrt{35}i}{2}\approx -3.5+2.958039892i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 8 by x+4.

$$x=x^{2}-4+8x+32-7$$

Add -4 and 32 to get 28.

$$x=x^{2}+28+8x-7$$

Subtract 7 from 28 to get 21.

$$x=x^{2}+21+8x$$

Subtract x^{2} from both sides.

$$x-x^{2}=21+8x$$

Subtract 21 from both sides.

$$x-x^{2}-21=8x$$

Subtract 8x from both sides.

$$x-x^{2}-21-8x=0$$

Combine x and -8x to get -7x.

$$-7x-x^{2}-21=0$$

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.

$$-x^{2}-7x-21=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -7 for b, and -21 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-1\right)\left(-21\right)}}{2\left(-1\right)}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-1\right)\left(-21\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-7\right)±\sqrt{49+4\left(-21\right)}}{2\left(-1\right)}$$

Multiply 4 times -21.

$$x=\frac{-\left(-7\right)±\sqrt{49-84}}{2\left(-1\right)}$$

Add 49 to -84.

$$x=\frac{-\left(-7\right)±\sqrt{-35}}{2\left(-1\right)}$$

Take the square root of -35.

$$x=\frac{-\left(-7\right)±\sqrt{35}i}{2\left(-1\right)}$$

The opposite of -7 is 7.

$$x=\frac{7±\sqrt{35}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{7±\sqrt{35}i}{-2}$$

Now solve the equation x=\frac{7±\sqrt{35}i}{-2} when ± is plus. Add 7 to i\sqrt{35}.

$$x=\frac{7+\sqrt{35}i}{-2}$$

Divide 7+i\sqrt{35} by -2.

$$x=\frac{-\sqrt{35}i-7}{2}$$

Now solve the equation x=\frac{7±\sqrt{35}i}{-2} when ± is minus. Subtract i\sqrt{35} from 7.

$$x=\frac{-\sqrt{35}i+7}{-2}$$

Divide 7-i\sqrt{35} by -2.

$$x=\frac{-7+\sqrt{35}i}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{35}i-7}{2} x=\frac{-7+\sqrt{35}i}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 8 by x+4.

$$x=x^{2}-4+8x+32-7$$

Add -4 and 32 to get 28.

$$x=x^{2}+28+8x-7$$

Subtract 7 from 28 to get 21.

$$x=x^{2}+21+8x$$

Subtract x^{2} from both sides.

$$x-x^{2}=21+8x$$

Subtract 8x from both sides.

$$x-x^{2}-8x=21$$

Combine x and -8x to get -7x.

$$-7x-x^{2}=21$$

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.

$$-x^{2}-7x=21$$

Divide both sides by -1.

$$\frac{-x^{2}-7x}{-1}=\frac{21}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{7}{-1}\right)x=\frac{21}{-1}$$

Divide -7 by -1.

$$x^{2}+7x=\frac{21}{-1}$$

Divide 21 by -1.

$$x^{2}+7x=-21$$

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.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=-21+\left(\frac{7}{2}\right)^{2}$$

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

$$x^{2}+7x+\frac{49}{4}=-21+\frac{49}{4}$$

Add -21 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=-\frac{35}{4}$$

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}.

$$\left(x+\frac{7}{2}\right)^{2}=-\frac{35}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{-\frac{35}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{\sqrt{35}i}{2} x+\frac{7}{2}=-\frac{\sqrt{35}i}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$x=\frac{-7+\sqrt{35}i}{2} x=\frac{-\sqrt{35}i-7}{2}$$