Solve a^2+6a+4(11-5a)+(11-5a)^2-21=0

$$a ^ { 2 } + 6 a + 4 ( 11 - 5 a ) + ( 11 - 5 a ) ^ { 2 } - 21 = 0$$

Solve for a

$$a=2$$

$$a = \frac{36}{13} = 2\frac{10}{13} \approx 2.769230769$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4 by 11-5a.

$$a^{2}+6a+44-20a+\left(11-5a\right)^{2}-21=0$$

Combine 6a and -20a to get -14a.

$$a^{2}-14a+44+\left(11-5a\right)^{2}-21=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(11-5a\right)^{2}.

$$a^{2}-14a+44+121-110a+25a^{2}-21=0$$

Add 44 and 121 to get 165.

$$a^{2}-14a+165-110a+25a^{2}-21=0$$

Combine -14a and -110a to get -124a.

$$a^{2}-124a+165+25a^{2}-21=0$$

Combine a^{2} and 25a^{2} to get 26a^{2}.

$$26a^{2}-124a+165-21=0$$

Subtract 21 from 165 to get 144.

$$26a^{2}-124a+144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 26 for a, -124 for b, and 144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-124\right)±\sqrt{\left(-124\right)^{2}-4\times 26\times 144}}{2\times 26}$$

Square -124.

$$a=\frac{-\left(-124\right)±\sqrt{15376-4\times 26\times 144}}{2\times 26}$$

Multiply -4 times 26.

$$a=\frac{-\left(-124\right)±\sqrt{15376-104\times 144}}{2\times 26}$$

Multiply -104 times 144.

$$a=\frac{-\left(-124\right)±\sqrt{15376-14976}}{2\times 26}$$

Add 15376 to -14976.

$$a=\frac{-\left(-124\right)±\sqrt{400}}{2\times 26}$$

Take the square root of 400.

$$a=\frac{-\left(-124\right)±20}{2\times 26}$$

The opposite of -124 is 124.

$$a=\frac{124±20}{2\times 26}$$

Multiply 2 times 26.

$$a=\frac{124±20}{52}$$

Now solve the equation a=\frac{124±20}{52} when ± is plus. Add 124 to 20.

$$a=\frac{144}{52}$$

Reduce the fraction \frac{144}{52} to lowest terms by extracting and canceling out 4.

$$a=\frac{36}{13}$$

Now solve the equation a=\frac{124±20}{52} when ± is minus. Subtract 20 from 124.

$$a=\frac{104}{52}$$

Divide 104 by 52.

$$a=2$$

The equation is now solved.

$$a=\frac{36}{13} a=2$$

Steps for Completing the Square

Use the distributive property to multiply 4 by 11-5a.

$$a^{2}+6a+44-20a+\left(11-5a\right)^{2}-21=0$$

Combine 6a and -20a to get -14a.

$$a^{2}-14a+44+\left(11-5a\right)^{2}-21=0$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(11-5a\right)^{2}.

$$a^{2}-14a+44+121-110a+25a^{2}-21=0$$

Add 44 and 121 to get 165.

$$a^{2}-14a+165-110a+25a^{2}-21=0$$

Combine -14a and -110a to get -124a.

$$a^{2}-124a+165+25a^{2}-21=0$$

Combine a^{2} and 25a^{2} to get 26a^{2}.

$$26a^{2}-124a+165-21=0$$

Subtract 21 from 165 to get 144.

$$26a^{2}-124a+144=0$$

Subtract 144 from both sides. Anything subtracted from zero gives its negation.

$$26a^{2}-124a=-144$$

Divide both sides by 26.

$$\frac{26a^{2}-124a}{26}=-\frac{144}{26}$$

Dividing by 26 undoes the multiplication by 26.

$$a^{2}+\left(-\frac{124}{26}\right)a=-\frac{144}{26}$$

Reduce the fraction \frac{-124}{26} to lowest terms by extracting and canceling out 2.

$$a^{2}-\frac{62}{13}a=-\frac{144}{26}$$

Reduce the fraction \frac{-144}{26} to lowest terms by extracting and canceling out 2.

$$a^{2}-\frac{62}{13}a=-\frac{72}{13}$$

Divide -\frac{62}{13}, the coefficient of the x term, by 2 to get -\frac{31}{13}. Then add the square of -\frac{31}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{62}{13}a+\left(-\frac{31}{13}\right)^{2}=-\frac{72}{13}+\left(-\frac{31}{13}\right)^{2}$$

Square -\frac{31}{13} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{62}{13}a+\frac{961}{169}=-\frac{72}{13}+\frac{961}{169}$$

Add -\frac{72}{13} to \frac{961}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{62}{13}a+\frac{961}{169}=\frac{25}{169}$$

Factor a^{2}-\frac{62}{13}a+\frac{961}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{31}{13}\right)^{2}=\frac{25}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{31}{13}\right)^{2}}=\sqrt{\frac{25}{169}}$$

Simplify.

$$a-\frac{31}{13}=\frac{5}{13} a-\frac{31}{13}=-\frac{5}{13}$$

Add \frac{31}{13} to both sides of the equation.

$$a=\frac{36}{13} a=2$$