Solve 3(5-x)*3(1-2x)=42

$$3 ( 5 - x ) \cdot 3 ( 1 - 2 x ) = 42$$

Solve for x

$$x=\frac{\sqrt{1065}}{12}+\frac{11}{4}\approx 5.469528145$$

$$x=-\frac{\sqrt{1065}}{12}+\frac{11}{4}\approx 0.030471855$$

Steps Using the Quadratic Formula

Multiply 3 and 3 to get 9.

$$9\left(5-x\right)\left(1-2x\right)=42$$

Use the distributive property to multiply 9 by 5-x.

$$\left(45-9x\right)\left(1-2x\right)=42$$

Use the distributive property to multiply 45-9x by 1-2x and combine like terms.

$$45-99x+18x^{2}=42$$

Subtract 42 from both sides.

$$45-99x+18x^{2}-42=0$$

Subtract 42 from 45 to get 3.

$$3-99x+18x^{2}=0$$

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.

$$18x^{2}-99x+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 18 for a, -99 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-99\right)±\sqrt{\left(-99\right)^{2}-4\times 18\times 3}}{2\times 18}$$

Square -99.

$$x=\frac{-\left(-99\right)±\sqrt{9801-4\times 18\times 3}}{2\times 18}$$

Multiply -4 times 18.

$$x=\frac{-\left(-99\right)±\sqrt{9801-72\times 3}}{2\times 18}$$

Multiply -72 times 3.

$$x=\frac{-\left(-99\right)±\sqrt{9801-216}}{2\times 18}$$

Add 9801 to -216.

$$x=\frac{-\left(-99\right)±\sqrt{9585}}{2\times 18}$$

Take the square root of 9585.

$$x=\frac{-\left(-99\right)±3\sqrt{1065}}{2\times 18}$$

The opposite of -99 is 99.

$$x=\frac{99±3\sqrt{1065}}{2\times 18}$$

Multiply 2 times 18.

$$x=\frac{99±3\sqrt{1065}}{36}$$

Now solve the equation x=\frac{99±3\sqrt{1065}}{36} when ± is plus. Add 99 to 3\sqrt{1065}.

$$x=\frac{3\sqrt{1065}+99}{36}$$

Divide 99+3\sqrt{1065} by 36.

$$x=\frac{\sqrt{1065}}{12}+\frac{11}{4}$$

Now solve the equation x=\frac{99±3\sqrt{1065}}{36} when ± is minus. Subtract 3\sqrt{1065} from 99.

$$x=\frac{99-3\sqrt{1065}}{36}$$

Divide 99-3\sqrt{1065} by 36.

$$x=-\frac{\sqrt{1065}}{12}+\frac{11}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{1065}}{12}+\frac{11}{4} x=-\frac{\sqrt{1065}}{12}+\frac{11}{4}$$

Steps for Completing the Square

Multiply 3 and 3 to get 9.

$$9\left(5-x\right)\left(1-2x\right)=42$$

Use the distributive property to multiply 9 by 5-x.

$$\left(45-9x\right)\left(1-2x\right)=42$$

Use the distributive property to multiply 45-9x by 1-2x and combine like terms.

$$45-99x+18x^{2}=42$$

Subtract 45 from both sides.

$$-99x+18x^{2}=42-45$$

Subtract 45 from 42 to get -3.

$$-99x+18x^{2}=-3$$

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.

$$18x^{2}-99x=-3$$

Divide both sides by 18.

$$\frac{18x^{2}-99x}{18}=-\frac{3}{18}$$

Dividing by 18 undoes the multiplication by 18.

$$x^{2}+\left(-\frac{99}{18}\right)x=-\frac{3}{18}$$

Reduce the fraction \frac{-99}{18} to lowest terms by extracting and canceling out 9.

$$x^{2}-\frac{11}{2}x=-\frac{3}{18}$$

Reduce the fraction \frac{-3}{18} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{11}{2}x=-\frac{1}{6}$$

Divide -\frac{11}{2}, the coefficient of the x term, by 2 to get -\frac{11}{4}. Then add the square of -\frac{11}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{2}x+\left(-\frac{11}{4}\right)^{2}=-\frac{1}{6}+\left(-\frac{11}{4}\right)^{2}$$

Square -\frac{11}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=-\frac{1}{6}+\frac{121}{16}$$

Add -\frac{1}{6} to \frac{121}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=\frac{355}{48}$$

Factor x^{2}-\frac{11}{2}x+\frac{121}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{4}\right)^{2}=\frac{355}{48}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{4}\right)^{2}}=\sqrt{\frac{355}{48}}$$

Simplify.

$$x-\frac{11}{4}=\frac{\sqrt{1065}}{12} x-\frac{11}{4}=-\frac{\sqrt{1065}}{12}$$

Add \frac{11}{4} to both sides of the equation.

$$x=\frac{\sqrt{1065}}{12}+\frac{11}{4} x=-\frac{\sqrt{1065}}{12}+\frac{11}{4}$$