Solve 21+64+82x+x^2+80=27+81x+602

$$21 + 64 + 82 x + x ^ { 2 } + 80 = 27 + 81 x + 602$$

Solve for x

$$x = \frac{\sqrt{1857} - 1}{2} \approx 21.046461426$$

$$x=\frac{-\sqrt{1857}-1}{2}\approx -22.046461426$$

Steps Using the Quadratic Formula

Add 21 and 64 to get 85.

$$85+82x+x^{2}+80=27+81x+602$$

Add 85 and 80 to get 165.

$$165+82x+x^{2}=27+81x+602$$

Add 27 and 602 to get 629.

$$165+82x+x^{2}=629+81x$$

Subtract 629 from both sides.

$$165+82x+x^{2}-629=81x$$

Subtract 629 from 165 to get -464.

$$-464+82x+x^{2}=81x$$

Subtract 81x from both sides.

$$-464+82x+x^{2}-81x=0$$

Combine 82x and -81x to get x.

$$-464+x+x^{2}=0$$

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.

$$x^{2}+x-464=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and -464 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-464\right)}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-464\right)}}{2}$$

Multiply -4 times -464.

$$x=\frac{-1±\sqrt{1+1856}}{2}$$

Add 1 to 1856.

$$x=\frac{-1±\sqrt{1857}}{2}$$

Now solve the equation x=\frac{-1±\sqrt{1857}}{2} when ± is plus. Add -1 to \sqrt{1857}.

$$x=\frac{\sqrt{1857}-1}{2}$$

Now solve the equation x=\frac{-1±\sqrt{1857}}{2} when ± is minus. Subtract \sqrt{1857} from -1.

$$x=\frac{-\sqrt{1857}-1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{1857}-1}{2} x=\frac{-\sqrt{1857}-1}{2}$$

Steps for Completing the Square

Add 21 and 64 to get 85.

$$85+82x+x^{2}+80=27+81x+602$$

Add 85 and 80 to get 165.

$$165+82x+x^{2}=27+81x+602$$

Add 27 and 602 to get 629.

$$165+82x+x^{2}=629+81x$$

Subtract 81x from both sides.

$$165+82x+x^{2}-81x=629$$

Combine 82x and -81x to get x.

$$165+x+x^{2}=629$$

Subtract 165 from both sides.

$$x+x^{2}=629-165$$

Subtract 165 from 629 to get 464.

$$x+x^{2}=464$$

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.

$$x^{2}+x=464$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=464+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=464+\frac{1}{4}$$

Add 464 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{1857}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{1857}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{1857}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{1857}}{2} x+\frac{1}{2}=-\frac{\sqrt{1857}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{1857}-1}{2} x=\frac{-\sqrt{1857}-1}{2}$$