Solve 1/2x+1/2=x^2+6x+5

$$\frac { 1 } { 2 } x + \frac { 1 } { 2 } = x ^ { 2 } + 6 x + 5$$

Solve for x

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

$$x=-1$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$\frac{1}{2}x+\frac{1}{2}-x^{2}=6x+5$$

Subtract 6x from both sides.

$$\frac{1}{2}x+\frac{1}{2}-x^{2}-6x=5$$

Combine \frac{1}{2}x and -6x to get -\frac{11}{2}x.

$$-\frac{11}{2}x+\frac{1}{2}-x^{2}=5$$

Subtract 5 from both sides.

$$-\frac{11}{2}x+\frac{1}{2}-x^{2}-5=0$$

Subtract 5 from \frac{1}{2} to get -\frac{9}{2}.

$$-\frac{11}{2}x-\frac{9}{2}-x^{2}=0$$

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.

$$-x^{2}-\frac{11}{2}x-\frac{9}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -\frac{11}{2} for b, and -\frac{9}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{11}{2}\right)±\sqrt{\left(-\frac{11}{2}\right)^{2}-4\left(-1\right)\left(-\frac{9}{2}\right)}}{2\left(-1\right)}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{11}{2}\right)±\sqrt{\frac{121}{4}-4\left(-1\right)\left(-\frac{9}{2}\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-\frac{11}{2}\right)±\sqrt{\frac{121}{4}+4\left(-\frac{9}{2}\right)}}{2\left(-1\right)}$$

Multiply 4 times -\frac{9}{2}.

$$x=\frac{-\left(-\frac{11}{2}\right)±\sqrt{\frac{121}{4}-18}}{2\left(-1\right)}$$

Add \frac{121}{4} to -18.

$$x=\frac{-\left(-\frac{11}{2}\right)±\sqrt{\frac{49}{4}}}{2\left(-1\right)}$$

Take the square root of \frac{49}{4}.

$$x=\frac{-\left(-\frac{11}{2}\right)±\frac{7}{2}}{2\left(-1\right)}$$

The opposite of -\frac{11}{2} is \frac{11}{2}.

$$x=\frac{\frac{11}{2}±\frac{7}{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{\frac{11}{2}±\frac{7}{2}}{-2}$$

Now solve the equation x=\frac{\frac{11}{2}±\frac{7}{2}}{-2} when ± is plus. Add \frac{11}{2} to \frac{7}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{9}{-2}$$

Divide 9 by -2.

$$x=-\frac{9}{2}$$

Now solve the equation x=\frac{\frac{11}{2}±\frac{7}{2}}{-2} when ± is minus. Subtract \frac{7}{2} from \frac{11}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{2}{-2}$$

Divide 2 by -2.

$$x=-1$$

The equation is now solved.

$$x=-\frac{9}{2} x=-1$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$\frac{1}{2}x+\frac{1}{2}-x^{2}=6x+5$$

Subtract 6x from both sides.

$$\frac{1}{2}x+\frac{1}{2}-x^{2}-6x=5$$

Combine \frac{1}{2}x and -6x to get -\frac{11}{2}x.

$$-\frac{11}{2}x+\frac{1}{2}-x^{2}=5$$

Subtract \frac{1}{2} from both sides.

$$-\frac{11}{2}x-x^{2}=5-\frac{1}{2}$$

Subtract \frac{1}{2} from 5 to get \frac{9}{2}.

$$-\frac{11}{2}x-x^{2}=\frac{9}{2}$$

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.

$$-x^{2}-\frac{11}{2}x=\frac{9}{2}$$

Divide both sides by -1.

$$\frac{-x^{2}-\frac{11}{2}x}{-1}=\frac{\frac{9}{2}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{\frac{11}{2}}{-1}\right)x=\frac{\frac{9}{2}}{-1}$$

Divide -\frac{11}{2} by -1.

$$x^{2}+\frac{11}{2}x=\frac{\frac{9}{2}}{-1}$$

Divide \frac{9}{2} by -1.

$$x^{2}+\frac{11}{2}x=-\frac{9}{2}$$

Divide \frac{11}{2}, the coefficient of the x term, by 2 to get \frac{11}{4}. Then add the square of \frac{11}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{11}{2}x+\left(\frac{11}{4}\right)^{2}=-\frac{9}{2}+\left(\frac{11}{4}\right)^{2}$$

Square \frac{11}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{11}{2}x+\frac{121}{16}=-\frac{9}{2}+\frac{121}{16}$$

Add -\frac{9}{2} to \frac{121}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{11}{2}x+\frac{121}{16}=\frac{49}{16}$$

Factor x^{2}+\frac{11}{2}x+\frac{121}{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}.

$$\left(x+\frac{11}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{11}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x+\frac{11}{4}=\frac{7}{4} x+\frac{11}{4}=-\frac{7}{4}$$

Subtract \frac{11}{4} from both sides of the equation.

$$x=-1 x=-\frac{9}{2}$$