Solve left(25+aright)left(525-aright)=10000

$$\left( 25+a \right) \left( 525-a \right) = 10000$$

Solve for a

$$a=25\sqrt{105}+250\approx 506.173769149$$

$$a=250-25\sqrt{105}\approx -6.173769149$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 25+a by 525-a and combine like terms.

$$13125+500a-a^{2}=10000$$

Subtract 10000 from both sides.

$$13125+500a-a^{2}-10000=0$$

Subtract 10000 from 13125 to get 3125.

$$3125+500a-a^{2}=0$$

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.

$$-a^{2}+500a+3125=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 500 for b, and 3125 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-500±\sqrt{500^{2}-4\left(-1\right)\times 3125}}{2\left(-1\right)}$$

Square 500.

$$a=\frac{-500±\sqrt{250000-4\left(-1\right)\times 3125}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-500±\sqrt{250000+4\times 3125}}{2\left(-1\right)}$$

Multiply 4 times 3125.

$$a=\frac{-500±\sqrt{250000+12500}}{2\left(-1\right)}$$

Add 250000 to 12500.

$$a=\frac{-500±\sqrt{262500}}{2\left(-1\right)}$$

Take the square root of 262500.

$$a=\frac{-500±50\sqrt{105}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{-500±50\sqrt{105}}{-2}$$

Now solve the equation a=\frac{-500±50\sqrt{105}}{-2} when ± is plus. Add -500 to 50\sqrt{105}.

$$a=\frac{50\sqrt{105}-500}{-2}$$

Divide -500+50\sqrt{105} by -2.

$$a=250-25\sqrt{105}$$

Now solve the equation a=\frac{-500±50\sqrt{105}}{-2} when ± is minus. Subtract 50\sqrt{105} from -500.

$$a=\frac{-50\sqrt{105}-500}{-2}$$

Divide -500-50\sqrt{105} by -2.

$$a=25\sqrt{105}+250$$

The equation is now solved.

$$a=250-25\sqrt{105} a=25\sqrt{105}+250$$

Steps for Completing the Square

Use the distributive property to multiply 25+a by 525-a and combine like terms.

$$13125+500a-a^{2}=10000$$

Subtract 13125 from both sides.

$$500a-a^{2}=10000-13125$$

Subtract 13125 from 10000 to get -3125.

$$500a-a^{2}=-3125$$

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.

$$-a^{2}+500a=-3125$$

Divide both sides by -1.

$$\frac{-a^{2}+500a}{-1}=-\frac{3125}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$a^{2}+\frac{500}{-1}a=-\frac{3125}{-1}$$

Divide 500 by -1.

$$a^{2}-500a=-\frac{3125}{-1}$$

Divide -3125 by -1.

$$a^{2}-500a=3125$$

Divide -500, the coefficient of the x term, by 2 to get -250. Then add the square of -250 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-500a+\left(-250\right)^{2}=3125+\left(-250\right)^{2}$$

Square -250.

$$a^{2}-500a+62500=3125+62500$$

Add 3125 to 62500.

$$a^{2}-500a+62500=65625$$

Factor a^{2}-500a+62500. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-250\right)^{2}=65625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-250\right)^{2}}=\sqrt{65625}$$

Simplify.

$$a-250=25\sqrt{105} a-250=-25\sqrt{105}$$

Add 250 to both sides of the equation.

$$a=25\sqrt{105}+250 a=250-25\sqrt{105}$$