Solve 3/4x-2(x+1)=-x^2

$$\frac { 3 } { 4 } x - 2 ( x + 1 ) = - x ^ { 2 }$$

Solve for x

$$x = \frac{3 \sqrt{17} + 5}{8} \approx 2.17116461$$

$$x=\frac{5-3\sqrt{17}}{8}\approx -0.92116461$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -2 by x+1.

$$\frac{3}{4}x-2x-2=-x^{2}$$

Combine \frac{3}{4}x and -2x to get -\frac{5}{4}x.

$$-\frac{5}{4}x-2=-x^{2}$$

Add x^{2} to both sides.

$$-\frac{5}{4}x-2+x^{2}=0$$

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.

$$x^{2}-\frac{5}{4}x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -\frac{5}{4} for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{5}{4}\right)±\sqrt{\left(-\frac{5}{4}\right)^{2}-4\left(-2\right)}}{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{5}{4}\right)±\sqrt{\frac{25}{16}-4\left(-2\right)}}{2}$$

Multiply -4 times -2.

$$x=\frac{-\left(-\frac{5}{4}\right)±\sqrt{\frac{25}{16}+8}}{2}$$

Add \frac{25}{16} to 8.

$$x=\frac{-\left(-\frac{5}{4}\right)±\sqrt{\frac{153}{16}}}{2}$$

Take the square root of \frac{153}{16}.

$$x=\frac{-\left(-\frac{5}{4}\right)±\frac{3\sqrt{17}}{4}}{2}$$

The opposite of -\frac{5}{4} is \frac{5}{4}.

$$x=\frac{\frac{5}{4}±\frac{3\sqrt{17}}{4}}{2}$$

Now solve the equation x=\frac{\frac{5}{4}±\frac{3\sqrt{17}}{4}}{2} when ± is plus. Add \frac{5}{4} to \frac{3\sqrt{17}}{4}.

$$x=\frac{3\sqrt{17}+5}{2\times 4}$$

Divide \frac{5+3\sqrt{17}}{4} by 2.

$$x=\frac{3\sqrt{17}+5}{8}$$

Now solve the equation x=\frac{\frac{5}{4}±\frac{3\sqrt{17}}{4}}{2} when ± is minus. Subtract \frac{3\sqrt{17}}{4} from \frac{5}{4}.

$$x=\frac{5-3\sqrt{17}}{2\times 4}$$

Divide \frac{5-3\sqrt{17}}{4} by 2.

$$x=\frac{5-3\sqrt{17}}{8}$$

The equation is now solved.

$$x=\frac{3\sqrt{17}+5}{8} x=\frac{5-3\sqrt{17}}{8}$$

Steps for Completing the Square

Use the distributive property to multiply -2 by x+1.

$$\frac{3}{4}x-2x-2=-x^{2}$$

Combine \frac{3}{4}x and -2x to get -\frac{5}{4}x.

$$-\frac{5}{4}x-2=-x^{2}$$

Add x^{2} to both sides.

$$-\frac{5}{4}x-2+x^{2}=0$$

Add 2 to both sides. Anything plus zero gives itself.

$$-\frac{5}{4}x+x^{2}=2$$

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.

$$x^{2}-\frac{5}{4}x=2$$

Divide -\frac{5}{4}, the coefficient of the x term, by 2 to get -\frac{5}{8}. Then add the square of -\frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{4}x+\left(-\frac{5}{8}\right)^{2}=2+\left(-\frac{5}{8}\right)^{2}$$

Square -\frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=2+\frac{25}{64}$$

Add 2 to \frac{25}{64}.

$$x^{2}-\frac{5}{4}x+\frac{25}{64}=\frac{153}{64}$$

Factor x^{2}-\frac{5}{4}x+\frac{25}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{8}\right)^{2}=\frac{153}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{8}\right)^{2}}=\sqrt{\frac{153}{64}}$$

Simplify.

$$x-\frac{5}{8}=\frac{3\sqrt{17}}{8} x-\frac{5}{8}=-\frac{3\sqrt{17}}{8}$$

Add \frac{5}{8} to both sides of the equation.

$$x=\frac{3\sqrt{17}+5}{8} x=\frac{5-3\sqrt{17}}{8}$$