Solve 1(x-3)-(x+2)+2x(x-1)-5=0

$$1 ( x - 3 ) - ( x + 2 ) + 2 x ( x - 1 ) - 5 = 0$$

Solve for x

$$x = \frac{\sqrt{21} + 1}{2} \approx 2.791287847$$

$$x=\frac{1-\sqrt{21}}{2}\approx -1.791287847$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 1 by x-3.

$$x-3-\left(x+2\right)+2x\left(x-1\right)-5=0$$

To find the opposite of x+2, find the opposite of each term.

$$x-3-x-2+2x\left(x-1\right)-5=0$$

Combine x and -x to get 0.

$$-3-2+2x\left(x-1\right)-5=0$$

Subtract 2 from -3 to get -5.

$$-5+2x\left(x-1\right)-5=0$$

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

$$-5+2x^{2}-2x-5=0$$

Subtract 5 from -5 to get -10.

$$-10+2x^{2}-2x=0$$

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.

$$2x^{2}-2x-10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -2 for b, and -10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\left(-10\right)}}{2\times 2}$$

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 2\left(-10\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-2\right)±\sqrt{4-8\left(-10\right)}}{2\times 2}$$

Multiply -8 times -10.

$$x=\frac{-\left(-2\right)±\sqrt{4+80}}{2\times 2}$$

Add 4 to 80.

$$x=\frac{-\left(-2\right)±\sqrt{84}}{2\times 2}$$

Take the square root of 84.

$$x=\frac{-\left(-2\right)±2\sqrt{21}}{2\times 2}$$

The opposite of -2 is 2.

$$x=\frac{2±2\sqrt{21}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{2±2\sqrt{21}}{4}$$

Now solve the equation x=\frac{2±2\sqrt{21}}{4} when ± is plus. Add 2 to 2\sqrt{21}.

$$x=\frac{2\sqrt{21}+2}{4}$$

Divide 2+2\sqrt{21} by 4.

$$x=\frac{\sqrt{21}+1}{2}$$

Now solve the equation x=\frac{2±2\sqrt{21}}{4} when ± is minus. Subtract 2\sqrt{21} from 2.

$$x=\frac{2-2\sqrt{21}}{4}$$

Divide 2-2\sqrt{21} by 4.

$$x=\frac{1-\sqrt{21}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{21}+1}{2} x=\frac{1-\sqrt{21}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 1 by x-3.

$$x-3-\left(x+2\right)+2x\left(x-1\right)-5=0$$

To find the opposite of x+2, find the opposite of each term.

$$x-3-x-2+2x\left(x-1\right)-5=0$$

Combine x and -x to get 0.

$$-3-2+2x\left(x-1\right)-5=0$$

Subtract 2 from -3 to get -5.

$$-5+2x\left(x-1\right)-5=0$$

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

$$-5+2x^{2}-2x-5=0$$

Subtract 5 from -5 to get -10.

$$-10+2x^{2}-2x=0$$

Add 10 to both sides. Anything plus zero gives itself.

$$2x^{2}-2x=10$$

Divide both sides by 2.

$$\frac{2x^{2}-2x}{2}=\frac{10}{2}$$

Dividing by 2 undoes the multiplication by 2.

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

Divide -2 by 2.

$$x^{2}-x=\frac{10}{2}$$

Divide 10 by 2.

$$x^{2}-x=5$$

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=5+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=5+\frac{1}{4}$$

Add 5 to \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{21}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{21}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{21}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{21}}{2} x-\frac{1}{2}=-\frac{\sqrt{21}}{2}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{21}+1}{2} x=\frac{1-\sqrt{21}}{2}$$