Solve x^2+27+2x+7x+x^2=47

$$x ^ { 2 } + 27 + 2 x + 7 x + x ^ { 2 } = 47$$

Solve for x

$$x = \frac{\sqrt{241} - 9}{4} \approx 1.631043674$$

$$x=\frac{-\sqrt{241}-9}{4}\approx -6.131043674$$

Steps Using the Quadratic Formula

Combine 2x and 7x to get 9x.

$$x^{2}+27+9x+x^{2}=47$$

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

$$2x^{2}+27+9x=47$$

Subtract 47 from both sides.

$$2x^{2}+27+9x-47=0$$

Subtract 47 from 27 to get -20.

$$2x^{2}-20+9x=0$$

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.

$$2x^{2}+9x-20=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 9 for b, and -20 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\times 2\left(-20\right)}}{2\times 2}$$

Square 9.

$$x=\frac{-9±\sqrt{81-4\times 2\left(-20\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-9±\sqrt{81-8\left(-20\right)}}{2\times 2}$$

Multiply -8 times -20.

$$x=\frac{-9±\sqrt{81+160}}{2\times 2}$$

Add 81 to 160.

$$x=\frac{-9±\sqrt{241}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-9±\sqrt{241}}{4}$$

Now solve the equation x=\frac{-9±\sqrt{241}}{4} when ± is plus. Add -9 to \sqrt{241}.

$$x=\frac{\sqrt{241}-9}{4}$$

Now solve the equation x=\frac{-9±\sqrt{241}}{4} when ± is minus. Subtract \sqrt{241} from -9.

$$x=\frac{-\sqrt{241}-9}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{241}-9}{4} x=\frac{-\sqrt{241}-9}{4}$$

Steps for Completing the Square

Combine 2x and 7x to get 9x.

$$x^{2}+27+9x+x^{2}=47$$

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

$$2x^{2}+27+9x=47$$

Subtract 27 from both sides.

$$2x^{2}+9x=47-27$$

Subtract 27 from 47 to get 20.

$$2x^{2}+9x=20$$

Divide both sides by 2.

$$\frac{2x^{2}+9x}{2}=\frac{20}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{9}{2}x=\frac{20}{2}$$

Divide 20 by 2.

$$x^{2}+\frac{9}{2}x=10$$

Divide \frac{9}{2}, the coefficient of the x term, by 2 to get \frac{9}{4}. Then add the square of \frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{9}{2}x+\left(\frac{9}{4}\right)^{2}=10+\left(\frac{9}{4}\right)^{2}$$

Square \frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=10+\frac{81}{16}$$

Add 10 to \frac{81}{16}.

$$x^{2}+\frac{9}{2}x+\frac{81}{16}=\frac{241}{16}$$

Factor x^{2}+\frac{9}{2}x+\frac{81}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{9}{4}\right)^{2}=\frac{241}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{9}{4}\right)^{2}}=\sqrt{\frac{241}{16}}$$

Simplify.

$$x+\frac{9}{4}=\frac{\sqrt{241}}{4} x+\frac{9}{4}=-\frac{\sqrt{241}}{4}$$

Subtract \frac{9}{4} from both sides of the equation.

$$x=\frac{\sqrt{241}-9}{4} x=\frac{-\sqrt{241}-9}{4}$$