Solve 2*21*52=(21-1)(20-lambda)(19-lambda)

$$2 \times 21 \times 52 = ( 21 - 1 ) ( 20 - \lambda ) ( 19 - \lambda )$$

Solve for λ

$$\lambda =-\frac{\sqrt{10945}}{10}+\frac{39}{2}\approx 9.038164597$$

$$\lambda =\frac{\sqrt{10945}}{10}+\frac{39}{2}\approx 29.961835403$$

Steps Using the Quadratic Formula

Multiply 2 and 21 to get 42.

$$42\times 52=\left(21-1\right)\left(20-\lambda \right)\left(19-\lambda \right)$$

Multiply 42 and 52 to get 2184.

$$2184=\left(21-1\right)\left(20-\lambda \right)\left(19-\lambda \right)$$

Subtract 1 from 21 to get 20.

$$2184=20\left(20-\lambda \right)\left(19-\lambda \right)$$

Use the distributive property to multiply 20 by 20-\lambda .

$$2184=\left(400-20\lambda \right)\left(19-\lambda \right)$$

Use the distributive property to multiply 400-20\lambda by 19-\lambda and combine like terms.

$$2184=7600-780\lambda +20\lambda ^{2}$$

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

$$7600-780\lambda +20\lambda ^{2}=2184$$

Subtract 2184 from both sides.

$$7600-780\lambda +20\lambda ^{2}-2184=0$$

Subtract 2184 from 7600 to get 5416.

$$5416-780\lambda +20\lambda ^{2}=0$$

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.

$$20\lambda ^{2}-780\lambda +5416=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 20 for a, -780 for b, and 5416 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\lambda =\frac{-\left(-780\right)±\sqrt{\left(-780\right)^{2}-4\times 20\times 5416}}{2\times 20}$$

Square -780.

$$\lambda =\frac{-\left(-780\right)±\sqrt{608400-4\times 20\times 5416}}{2\times 20}$$

Multiply -4 times 20.

$$\lambda =\frac{-\left(-780\right)±\sqrt{608400-80\times 5416}}{2\times 20}$$

Multiply -80 times 5416.

$$\lambda =\frac{-\left(-780\right)±\sqrt{608400-433280}}{2\times 20}$$

Add 608400 to -433280.

$$\lambda =\frac{-\left(-780\right)±\sqrt{175120}}{2\times 20}$$

Take the square root of 175120.

$$\lambda =\frac{-\left(-780\right)±4\sqrt{10945}}{2\times 20}$$

The opposite of -780 is 780.

$$\lambda =\frac{780±4\sqrt{10945}}{2\times 20}$$

Multiply 2 times 20.

$$\lambda =\frac{780±4\sqrt{10945}}{40}$$

Now solve the equation \lambda =\frac{780±4\sqrt{10945}}{40} when ± is plus. Add 780 to 4\sqrt{10945}.

$$\lambda =\frac{4\sqrt{10945}+780}{40}$$

Divide 780+4\sqrt{10945} by 40.

$$\lambda =\frac{\sqrt{10945}}{10}+\frac{39}{2}$$

Now solve the equation \lambda =\frac{780±4\sqrt{10945}}{40} when ± is minus. Subtract 4\sqrt{10945} from 780.

$$\lambda =\frac{780-4\sqrt{10945}}{40}$$

Divide 780-4\sqrt{10945} by 40.

$$\lambda =-\frac{\sqrt{10945}}{10}+\frac{39}{2}$$

The equation is now solved.

$$\lambda =\frac{\sqrt{10945}}{10}+\frac{39}{2} \lambda =-\frac{\sqrt{10945}}{10}+\frac{39}{2}$$

Steps for Completing the Square

Multiply 2 and 21 to get 42.

$$42\times 52=\left(21-1\right)\left(20-\lambda \right)\left(19-\lambda \right)$$

Multiply 42 and 52 to get 2184.

$$2184=\left(21-1\right)\left(20-\lambda \right)\left(19-\lambda \right)$$

Subtract 1 from 21 to get 20.

$$2184=20\left(20-\lambda \right)\left(19-\lambda \right)$$

Use the distributive property to multiply 20 by 20-\lambda .

$$2184=\left(400-20\lambda \right)\left(19-\lambda \right)$$

Use the distributive property to multiply 400-20\lambda by 19-\lambda and combine like terms.

$$2184=7600-780\lambda +20\lambda ^{2}$$

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

$$7600-780\lambda +20\lambda ^{2}=2184$$

Subtract 7600 from both sides.

$$-780\lambda +20\lambda ^{2}=2184-7600$$

Subtract 7600 from 2184 to get -5416.

$$-780\lambda +20\lambda ^{2}=-5416$$

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.

$$20\lambda ^{2}-780\lambda =-5416$$

Divide both sides by 20.

$$\frac{20\lambda ^{2}-780\lambda }{20}=-\frac{5416}{20}$$

Dividing by 20 undoes the multiplication by 20.

$$\lambda ^{2}+\left(-\frac{780}{20}\right)\lambda =-\frac{5416}{20}$$

Divide -780 by 20.

$$\lambda ^{2}-39\lambda =-\frac{5416}{20}$$

Reduce the fraction \frac{-5416}{20} to lowest terms by extracting and canceling out 4.

$$\lambda ^{2}-39\lambda =-\frac{1354}{5}$$

Divide -39, the coefficient of the x term, by 2 to get -\frac{39}{2}. Then add the square of -\frac{39}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\lambda ^{2}-39\lambda +\left(-\frac{39}{2}\right)^{2}=-\frac{1354}{5}+\left(-\frac{39}{2}\right)^{2}$$

Square -\frac{39}{2} by squaring both the numerator and the denominator of the fraction.

$$\lambda ^{2}-39\lambda +\frac{1521}{4}=-\frac{1354}{5}+\frac{1521}{4}$$

Add -\frac{1354}{5} to \frac{1521}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$\lambda ^{2}-39\lambda +\frac{1521}{4}=\frac{2189}{20}$$

Factor \lambda ^{2}-39\lambda +\frac{1521}{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}.

$$\left(\lambda -\frac{39}{2}\right)^{2}=\frac{2189}{20}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\lambda -\frac{39}{2}\right)^{2}}=\sqrt{\frac{2189}{20}}$$

Simplify.

$$\lambda -\frac{39}{2}=\frac{\sqrt{10945}}{10} \lambda -\frac{39}{2}=-\frac{\sqrt{10945}}{10}$$

Add \frac{39}{2} to both sides of the equation.

$$\lambda =\frac{\sqrt{10945}}{10}+\frac{39}{2} \lambda =-\frac{\sqrt{10945}}{10}+\frac{39}{2}$$