Solve 4psi^2-2=3psi(psi+1)+6

$$4 \psi ^ { 2 } - 2 = 3 \psi ( \psi + 1 ) + 6$$

Solve for ψ

$$\psi =\frac{\sqrt{41}+3}{2}\approx 4.701562119$$

$$\psi =\frac{3-\sqrt{41}}{2}\approx -1.701562119$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3\psi by \psi +1.

$$4\psi ^{2}-2=3\psi ^{2}+3\psi +6$$

Subtract 3\psi ^{2} from both sides.

$$4\psi ^{2}-2-3\psi ^{2}=3\psi +6$$

Combine 4\psi ^{2} and -3\psi ^{2} to get \psi ^{2}.

$$\psi ^{2}-2=3\psi +6$$

Subtract 3\psi from both sides.

$$\psi ^{2}-2-3\psi =6$$

Subtract 6 from both sides.

$$\psi ^{2}-2-3\psi -6=0$$

Subtract 6 from -2 to get -8.

$$\psi ^{2}-8-3\psi =0$$

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.

$$\psi ^{2}-3\psi -8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\psi =\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-8\right)}}{2}$$

Square -3.

$$\psi =\frac{-\left(-3\right)±\sqrt{9-4\left(-8\right)}}{2}$$

Multiply -4 times -8.

$$\psi =\frac{-\left(-3\right)±\sqrt{9+32}}{2}$$

Add 9 to 32.

$$\psi =\frac{-\left(-3\right)±\sqrt{41}}{2}$$

The opposite of -3 is 3.

$$\psi =\frac{3±\sqrt{41}}{2}$$

Now solve the equation \psi =\frac{3±\sqrt{41}}{2} when ± is plus. Add 3 to \sqrt{41}.

$$\psi =\frac{\sqrt{41}+3}{2}$$

Now solve the equation \psi =\frac{3±\sqrt{41}}{2} when ± is minus. Subtract \sqrt{41} from 3.

$$\psi =\frac{3-\sqrt{41}}{2}$$

The equation is now solved.

$$\psi =\frac{\sqrt{41}+3}{2} \psi =\frac{3-\sqrt{41}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 3\psi by \psi +1.

$$4\psi ^{2}-2=3\psi ^{2}+3\psi +6$$

Subtract 3\psi ^{2} from both sides.

$$4\psi ^{2}-2-3\psi ^{2}=3\psi +6$$

Combine 4\psi ^{2} and -3\psi ^{2} to get \psi ^{2}.

$$\psi ^{2}-2=3\psi +6$$

Subtract 3\psi from both sides.

$$\psi ^{2}-2-3\psi =6$$

Add 2 to both sides.

$$\psi ^{2}-3\psi =6+2$$

Add 6 and 2 to get 8.

$$\psi ^{2}-3\psi =8$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\psi ^{2}-3\psi +\left(-\frac{3}{2}\right)^{2}=8+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$\psi ^{2}-3\psi +\frac{9}{4}=8+\frac{9}{4}$$

Add 8 to \frac{9}{4}.

$$\psi ^{2}-3\psi +\frac{9}{4}=\frac{41}{4}$$

Factor \psi ^{2}-3\psi +\frac{9}{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}.

$$\left(\psi -\frac{3}{2}\right)^{2}=\frac{41}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\psi -\frac{3}{2}\right)^{2}}=\sqrt{\frac{41}{4}}$$

Simplify.

$$\psi -\frac{3}{2}=\frac{\sqrt{41}}{2} \psi -\frac{3}{2}=-\frac{\sqrt{41}}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$\psi =\frac{\sqrt{41}+3}{2} \psi =\frac{3-\sqrt{41}}{2}$$