Solve (a-40)(1000-10a)=8000

$$( a - 40 ) ( 1000 - 10 a ) = 8000$$

Solve for a

$$a=60$$

$$a=80$$

Steps Using the Quadratic Formula

Use the distributive property to multiply a-40 by 1000-10a and combine like terms.

$$1400a-10a^{2}-40000=8000$$

Subtract 8000 from both sides.

$$1400a-10a^{2}-40000-8000=0$$

Subtract 8000 from -40000 to get -48000.

$$1400a-10a^{2}-48000=0$$

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.

$$-10a^{2}+1400a-48000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -10 for a, 1400 for b, and -48000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-1400±\sqrt{1400^{2}-4\left(-10\right)\left(-48000\right)}}{2\left(-10\right)}$$

Square 1400.

$$a=\frac{-1400±\sqrt{1960000-4\left(-10\right)\left(-48000\right)}}{2\left(-10\right)}$$

Multiply -4 times -10.

$$a=\frac{-1400±\sqrt{1960000+40\left(-48000\right)}}{2\left(-10\right)}$$

Multiply 40 times -48000.

$$a=\frac{-1400±\sqrt{1960000-1920000}}{2\left(-10\right)}$$

Add 1960000 to -1920000.

$$a=\frac{-1400±\sqrt{40000}}{2\left(-10\right)}$$

Take the square root of 40000.

$$a=\frac{-1400±200}{2\left(-10\right)}$$

Multiply 2 times -10.

$$a=\frac{-1400±200}{-20}$$

Now solve the equation a=\frac{-1400±200}{-20} when ± is plus. Add -1400 to 200.

$$a=-\frac{1200}{-20}$$

Divide -1200 by -20.

$$a=60$$

Now solve the equation a=\frac{-1400±200}{-20} when ± is minus. Subtract 200 from -1400.

$$a=-\frac{1600}{-20}$$

Divide -1600 by -20.

$$a=80$$

The equation is now solved.

$$a=60 a=80$$

Steps for Completing the Square

Use the distributive property to multiply a-40 by 1000-10a and combine like terms.

$$1400a-10a^{2}-40000=8000$$

Add 40000 to both sides.

$$1400a-10a^{2}=8000+40000$$

Add 8000 and 40000 to get 48000.

$$1400a-10a^{2}=48000$$

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.

$$-10a^{2}+1400a=48000$$

Divide both sides by -10.

$$\frac{-10a^{2}+1400a}{-10}=\frac{48000}{-10}$$

Dividing by -10 undoes the multiplication by -10.

$$a^{2}+\frac{1400}{-10}a=\frac{48000}{-10}$$

Divide 1400 by -10.

$$a^{2}-140a=\frac{48000}{-10}$$

Divide 48000 by -10.

$$a^{2}-140a=-4800$$

Divide -140, the coefficient of the x term, by 2 to get -70. Then add the square of -70 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-140a+\left(-70\right)^{2}=-4800+\left(-70\right)^{2}$$

Square -70.

$$a^{2}-140a+4900=-4800+4900$$

Add -4800 to 4900.

$$a^{2}-140a+4900=100$$

Factor a^{2}-140a+4900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-70\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-70\right)^{2}}=\sqrt{100}$$

Simplify.

$$a-70=10 a-70=-10$$

Add 70 to both sides of the equation.

$$a=80 a=60$$