Solve left(20x+40right)left(60-45-xright)=1265

$$\left( 20x+40 \right) \left( 60-45-x \right) = 1265$$

Solve for x

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

$$x = \frac{19}{2} = 9\frac{1}{2} = 9.5$$

Steps Using the Quadratic Formula

Subtract 45 from 60 to get 15.

$$\left(20x+40\right)\left(15-x\right)=1265$$

Use the distributive property to multiply 20x+40 by 15-x and combine like terms.

$$260x-20x^{2}+600=1265$$

Subtract 1265 from both sides.

$$260x-20x^{2}+600-1265=0$$

Subtract 1265 from 600 to get -665.

$$260x-20x^{2}-665=0$$

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.

$$-20x^{2}+260x-665=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -20 for a, 260 for b, and -665 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-260±\sqrt{260^{2}-4\left(-20\right)\left(-665\right)}}{2\left(-20\right)}$$

Square 260.

$$x=\frac{-260±\sqrt{67600-4\left(-20\right)\left(-665\right)}}{2\left(-20\right)}$$

Multiply -4 times -20.

$$x=\frac{-260±\sqrt{67600+80\left(-665\right)}}{2\left(-20\right)}$$

Multiply 80 times -665.

$$x=\frac{-260±\sqrt{67600-53200}}{2\left(-20\right)}$$

Add 67600 to -53200.

$$x=\frac{-260±\sqrt{14400}}{2\left(-20\right)}$$

Take the square root of 14400.

$$x=\frac{-260±120}{2\left(-20\right)}$$

Multiply 2 times -20.

$$x=\frac{-260±120}{-40}$$

Now solve the equation x=\frac{-260±120}{-40} when ± is plus. Add -260 to 120.

$$x=-\frac{140}{-40}$$

Reduce the fraction \frac{-140}{-40} to lowest terms by extracting and canceling out 20.

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

Now solve the equation x=\frac{-260±120}{-40} when ± is minus. Subtract 120 from -260.

$$x=-\frac{380}{-40}$$

Reduce the fraction \frac{-380}{-40} to lowest terms by extracting and canceling out 20.

$$x=\frac{19}{2}$$

The equation is now solved.

$$x=\frac{7}{2} x=\frac{19}{2}$$

Steps for Completing the Square

Subtract 45 from 60 to get 15.

$$\left(20x+40\right)\left(15-x\right)=1265$$

Use the distributive property to multiply 20x+40 by 15-x and combine like terms.

$$260x-20x^{2}+600=1265$$

Subtract 600 from both sides.

$$260x-20x^{2}=1265-600$$

Subtract 600 from 1265 to get 665.

$$260x-20x^{2}=665$$

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.

$$-20x^{2}+260x=665$$

Divide both sides by -20.

$$\frac{-20x^{2}+260x}{-20}=\frac{665}{-20}$$

Dividing by -20 undoes the multiplication by -20.

$$x^{2}+\frac{260}{-20}x=\frac{665}{-20}$$

Divide 260 by -20.

$$x^{2}-13x=\frac{665}{-20}$$

Reduce the fraction \frac{665}{-20} to lowest terms by extracting and canceling out 5.

$$x^{2}-13x=-\frac{133}{4}$$

Divide -13, the coefficient of the x term, by 2 to get -\frac{13}{2}. Then add the square of -\frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=-\frac{133}{4}+\left(-\frac{13}{2}\right)^{2}$$

Square -\frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-13x+\frac{169}{4}=\frac{-133+169}{4}$$

Add -\frac{133}{4} to \frac{169}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-13x+\frac{169}{4}=9$$

Factor x^{2}-13x+\frac{169}{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}.

$$\left(x-\frac{13}{2}\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{9}$$

Simplify.

$$x-\frac{13}{2}=3 x-\frac{13}{2}=-3$$

Add \frac{13}{2} to both sides of the equation.

$$x=\frac{19}{2} x=\frac{7}{2}$$