Solve 2(a-5)(a-5)=21a

$$2 ( a - 5 ) ( a - 5 ) = 21 a$$

Solve for a

$$a = \frac{\sqrt{1281} + 41}{4} \approx 19.197765084$$

$$a = \frac{41 - \sqrt{1281}}{4} \approx 1.302234916$$

Steps Using the Quadratic Formula

Multiply a-5 and a-5 to get \left(a-5\right)^{2}.

$$2\left(a-5\right)^{2}=21a$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(a-5\right)^{2}.

$$2\left(a^{2}-10a+25\right)=21a$$

Use the distributive property to multiply 2 by a^{2}-10a+25.

$$2a^{2}-20a+50=21a$$

Subtract 21a from both sides.

$$2a^{2}-20a+50-21a=0$$

Combine -20a and -21a to get -41a.

$$2a^{2}-41a+50=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -41 for b, and 50 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-41\right)±\sqrt{\left(-41\right)^{2}-4\times 2\times 50}}{2\times 2}$$

Square -41.

$$a=\frac{-\left(-41\right)±\sqrt{1681-4\times 2\times 50}}{2\times 2}$$

Multiply -4 times 2.

$$a=\frac{-\left(-41\right)±\sqrt{1681-8\times 50}}{2\times 2}$$

Multiply -8 times 50.

$$a=\frac{-\left(-41\right)±\sqrt{1681-400}}{2\times 2}$$

Add 1681 to -400.

$$a=\frac{-\left(-41\right)±\sqrt{1281}}{2\times 2}$$

The opposite of -41 is 41.

$$a=\frac{41±\sqrt{1281}}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{41±\sqrt{1281}}{4}$$

Now solve the equation a=\frac{41±\sqrt{1281}}{4} when ± is plus. Add 41 to \sqrt{1281}.

$$a=\frac{\sqrt{1281}+41}{4}$$

Now solve the equation a=\frac{41±\sqrt{1281}}{4} when ± is minus. Subtract \sqrt{1281} from 41.

$$a=\frac{41-\sqrt{1281}}{4}$$

The equation is now solved.

$$a=\frac{\sqrt{1281}+41}{4} a=\frac{41-\sqrt{1281}}{4}$$

Steps for Completing the Square

Multiply a-5 and a-5 to get \left(a-5\right)^{2}.

$$2\left(a-5\right)^{2}=21a$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(a-5\right)^{2}.

$$2\left(a^{2}-10a+25\right)=21a$$

Use the distributive property to multiply 2 by a^{2}-10a+25.

$$2a^{2}-20a+50=21a$$

Subtract 21a from both sides.

$$2a^{2}-20a+50-21a=0$$

Combine -20a and -21a to get -41a.

$$2a^{2}-41a+50=0$$

Subtract 50 from both sides. Anything subtracted from zero gives its negation.

$$2a^{2}-41a=-50$$

Divide both sides by 2.

$$\frac{2a^{2}-41a}{2}=-\frac{50}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$a^{2}-\frac{41}{2}a=-\frac{50}{2}$$

Divide -50 by 2.

$$a^{2}-\frac{41}{2}a=-25$$

Divide -\frac{41}{2}, the coefficient of the x term, by 2 to get -\frac{41}{4}. Then add the square of -\frac{41}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{41}{2}a+\left(-\frac{41}{4}\right)^{2}=-25+\left(-\frac{41}{4}\right)^{2}$$

Square -\frac{41}{4} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{41}{2}a+\frac{1681}{16}=-25+\frac{1681}{16}$$

Add -25 to \frac{1681}{16}.

$$a^{2}-\frac{41}{2}a+\frac{1681}{16}=\frac{1281}{16}$$

Factor a^{2}-\frac{41}{2}a+\frac{1681}{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}.

$$\left(a-\frac{41}{4}\right)^{2}=\frac{1281}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{41}{4}\right)^{2}}=\sqrt{\frac{1281}{16}}$$

Simplify.

$$a-\frac{41}{4}=\frac{\sqrt{1281}}{4} a-\frac{41}{4}=-\frac{\sqrt{1281}}{4}$$

Add \frac{41}{4} to both sides of the equation.

$$a=\frac{\sqrt{1281}+41}{4} a=\frac{41-\sqrt{1281}}{4}$$