Solve 0.5*(2-x)(2-1.5x)=5

$$0.5 \times (2-x)(2-1.5x)=5$$

Solve for x

$$x = \frac{\sqrt{61} + 5}{3} \approx 4.270083225$$

$$x=\frac{5-\sqrt{61}}{3}\approx -0.936749892$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 0.5 by 2-x.

$$\left(1-0.5x\right)\left(2-1.5x\right)=5$$

Use the distributive property to multiply 1-0.5x by 2-1.5x and combine like terms.

$$2-2.5x+0.75x^{2}=5$$

Subtract 5 from both sides.

$$2-2.5x+0.75x^{2}-5=0$$

Subtract 5 from 2 to get -3.

$$-3-2.5x+0.75x^{2}=0$$

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.

$$0.75x^{2}-2.5x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.75 for a, -2.5 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2.5\right)±\sqrt{\left(-2.5\right)^{2}-4\times 0.75\left(-3\right)}}{2\times 0.75}$$

Square -2.5 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-2.5\right)±\sqrt{6.25-4\times 0.75\left(-3\right)}}{2\times 0.75}$$

Multiply -4 times 0.75.

$$x=\frac{-\left(-2.5\right)±\sqrt{6.25-3\left(-3\right)}}{2\times 0.75}$$

Multiply -3 times -3.

$$x=\frac{-\left(-2.5\right)±\sqrt{6.25+9}}{2\times 0.75}$$

Add 6.25 to 9.

$$x=\frac{-\left(-2.5\right)±\sqrt{15.25}}{2\times 0.75}$$

Take the square root of 15.25.

$$x=\frac{-\left(-2.5\right)±\frac{\sqrt{61}}{2}}{2\times 0.75}$$

The opposite of -2.5 is 2.5.

$$x=\frac{2.5±\frac{\sqrt{61}}{2}}{2\times 0.75}$$

Multiply 2 times 0.75.

$$x=\frac{2.5±\frac{\sqrt{61}}{2}}{1.5}$$

Now solve the equation x=\frac{2.5±\frac{\sqrt{61}}{2}}{1.5} when ± is plus. Add 2.5 to \frac{\sqrt{61}}{2}.

$$x=\frac{\sqrt{61}+5}{1.5\times 2}$$

Divide \frac{5+\sqrt{61}}{2} by 1.5 by multiplying \frac{5+\sqrt{61}}{2} by the reciprocal of 1.5.

$$x=\frac{\sqrt{61}+5}{3}$$

Now solve the equation x=\frac{2.5±\frac{\sqrt{61}}{2}}{1.5} when ± is minus. Subtract \frac{\sqrt{61}}{2} from 2.5.

$$x=\frac{5-\sqrt{61}}{1.5\times 2}$$

Divide \frac{5-\sqrt{61}}{2} by 1.5 by multiplying \frac{5-\sqrt{61}}{2} by the reciprocal of 1.5.

$$x=\frac{5-\sqrt{61}}{3}$$

The equation is now solved.

$$x=\frac{\sqrt{61}+5}{3} x=\frac{5-\sqrt{61}}{3}$$

Steps for Completing the Square

Use the distributive property to multiply 0.5 by 2-x.

$$\left(1-0.5x\right)\left(2-1.5x\right)=5$$

Use the distributive property to multiply 1-0.5x by 2-1.5x and combine like terms.

$$2-2.5x+0.75x^{2}=5$$

Subtract 2 from both sides.

$$-2.5x+0.75x^{2}=5-2$$

Subtract 2 from 5 to get 3.

$$-2.5x+0.75x^{2}=3$$

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.

$$0.75x^{2}-2.5x=3$$

Divide both sides of the equation by 0.75, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.75x^{2}-2.5x}{0.75}=\frac{3}{0.75}$$

Dividing by 0.75 undoes the multiplication by 0.75.

$$x^{2}+\left(-\frac{2.5}{0.75}\right)x=\frac{3}{0.75}$$

Divide -2.5 by 0.75 by multiplying -2.5 by the reciprocal of 0.75.

$$x^{2}-\frac{10}{3}x=\frac{3}{0.75}$$

Divide 3 by 0.75 by multiplying 3 by the reciprocal of 0.75.

$$x^{2}-\frac{10}{3}x=4$$

Divide -\frac{10}{3}, the coefficient of the x term, by 2 to get -\frac{5}{3}. Then add the square of -\frac{5}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{10}{3}x+\left(-\frac{5}{3}\right)^{2}=4+\left(-\frac{5}{3}\right)^{2}$$

Square -\frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{10}{3}x+\frac{25}{9}=4+\frac{25}{9}$$

Add 4 to \frac{25}{9}.

$$x^{2}-\frac{10}{3}x+\frac{25}{9}=\frac{61}{9}$$

Factor x^{2}-\frac{10}{3}x+\frac{25}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{3}\right)^{2}=\frac{61}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{3}\right)^{2}}=\sqrt{\frac{61}{9}}$$

Simplify.

$$x-\frac{5}{3}=\frac{\sqrt{61}}{3} x-\frac{5}{3}=-\frac{\sqrt{61}}{3}$$

Add \frac{5}{3} to both sides of the equation.

$$x=\frac{\sqrt{61}+5}{3} x=\frac{5-\sqrt{61}}{3}$$