Solve 30*0.6=30-0.5*(5-x)(12-2x)

$$30 \times 0.6=30-0.5 \times (5-x)(12-2x)$$

Solve for x

$$x=2$$

$$x=9$$

Steps Using the Quadratic Formula

Multiply 30 and 0.6 to get 18.

$$18=30-0.5\left(5-x\right)\left(12-2x\right)$$

Swap sides so that all variable terms are on the left hand side.

$$30-0.5\left(5-x\right)\left(12-2x\right)=18$$

Subtract 18 from both sides.

$$30-0.5\left(5-x\right)\left(12-2x\right)-18=0$$

Use the distributive property to multiply -0.5 by 5-x.

$$30+\left(-2.5+0.5x\right)\left(12-2x\right)-18=0$$

Use the distributive property to multiply -2.5+0.5x by 12-2x and combine like terms.

$$30-30+11x-x^{2}-18=0$$

Subtract 30 from 30 to get 0.

$$11x-x^{2}-18=0$$

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.

$$-x^{2}+11x-18=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 11 for b, and -18 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 11.

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

Multiply -4 times -1.

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

Multiply 4 times -18.

$$x=\frac{-11±\sqrt{121-72}}{2\left(-1\right)}$$

Add 121 to -72.

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

Take the square root of 49.

$$x=\frac{-11±7}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-11±7}{-2}$$

Now solve the equation x=\frac{-11±7}{-2} when ± is plus. Add -11 to 7.

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

Divide -4 by -2.

$$x=2$$

Now solve the equation x=\frac{-11±7}{-2} when ± is minus. Subtract 7 from -11.

$$x=-\frac{18}{-2}$$

Divide -18 by -2.

$$x=9$$

The equation is now solved.

$$x=2 x=9$$

Steps for Completing the Square

Multiply 30 and 0.6 to get 18.

$$18=30-0.5\left(5-x\right)\left(12-2x\right)$$

Swap sides so that all variable terms are on the left hand side.

$$30-0.5\left(5-x\right)\left(12-2x\right)=18$$

Use the distributive property to multiply -0.5 by 5-x.

$$30+\left(-2.5+0.5x\right)\left(12-2x\right)=18$$

Use the distributive property to multiply -2.5+0.5x by 12-2x and combine like terms.

$$30-30+11x-x^{2}=18$$

Subtract 30 from 30 to get 0.

$$11x-x^{2}=18$$

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.

$$-x^{2}+11x=18$$

Divide both sides by -1.

$$\frac{-x^{2}+11x}{-1}=\frac{18}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{11}{-1}x=\frac{18}{-1}$$

Divide 11 by -1.

$$x^{2}-11x=\frac{18}{-1}$$

Divide 18 by -1.

$$x^{2}-11x=-18$$

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

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-18+\left(-\frac{11}{2}\right)^{2}$$

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

$$x^{2}-11x+\frac{121}{4}=-18+\frac{121}{4}$$

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

$$x^{2}-11x+\frac{121}{4}=\frac{49}{4}$$

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

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add \frac{11}{2} to both sides of the equation.

$$x=9 x=2$$