Solve a^2+5a+3+7a+6a^2-4=6

$$a ^ { 2 } + 5 a + 3 + 7 a + 6 a ^ { 2 } - 4 = 6$$

Solve for a

$$a=\frac{\sqrt{85}-6}{7}\approx 0.459934922$$

$$a=\frac{-\sqrt{85}-6}{7}\approx -2.174220637$$

Steps Using the Quadratic Formula

Combine 5a and 7a to get 12a.

$$a^{2}+12a+3+6a^{2}-4=6$$

Combine a^{2} and 6a^{2} to get 7a^{2}.

$$7a^{2}+12a+3-4=6$$

Subtract 4 from 3 to get -1.

$$7a^{2}+12a-1=6$$

Subtract 6 from both sides.

$$7a^{2}+12a-1-6=0$$

Subtract 6 from -1 to get -7.

$$7a^{2}+12a-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, 12 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-12±\sqrt{12^{2}-4\times 7\left(-7\right)}}{2\times 7}$$

Square 12.

$$a=\frac{-12±\sqrt{144-4\times 7\left(-7\right)}}{2\times 7}$$

Multiply -4 times 7.

$$a=\frac{-12±\sqrt{144-28\left(-7\right)}}{2\times 7}$$

Multiply -28 times -7.

$$a=\frac{-12±\sqrt{144+196}}{2\times 7}$$

Add 144 to 196.

$$a=\frac{-12±\sqrt{340}}{2\times 7}$$

Take the square root of 340.

$$a=\frac{-12±2\sqrt{85}}{2\times 7}$$

Multiply 2 times 7.

$$a=\frac{-12±2\sqrt{85}}{14}$$

Now solve the equation a=\frac{-12±2\sqrt{85}}{14} when ± is plus. Add -12 to 2\sqrt{85}.

$$a=\frac{2\sqrt{85}-12}{14}$$

Divide -12+2\sqrt{85} by 14.

$$a=\frac{\sqrt{85}-6}{7}$$

Now solve the equation a=\frac{-12±2\sqrt{85}}{14} when ± is minus. Subtract 2\sqrt{85} from -12.

$$a=\frac{-2\sqrt{85}-12}{14}$$

Divide -12-2\sqrt{85} by 14.

$$a=\frac{-\sqrt{85}-6}{7}$$

The equation is now solved.

$$a=\frac{\sqrt{85}-6}{7} a=\frac{-\sqrt{85}-6}{7}$$

Steps for Completing the Square

Combine 5a and 7a to get 12a.

$$a^{2}+12a+3+6a^{2}-4=6$$

Combine a^{2} and 6a^{2} to get 7a^{2}.

$$7a^{2}+12a+3-4=6$$

Subtract 4 from 3 to get -1.

$$7a^{2}+12a-1=6$$

Add 1 to both sides.

$$7a^{2}+12a=6+1$$

Add 6 and 1 to get 7.

$$7a^{2}+12a=7$$

Divide both sides by 7.

$$\frac{7a^{2}+12a}{7}=\frac{7}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$a^{2}+\frac{12}{7}a=\frac{7}{7}$$

Divide 7 by 7.

$$a^{2}+\frac{12}{7}a=1$$

Divide \frac{12}{7}, the coefficient of the x term, by 2 to get \frac{6}{7}. Then add the square of \frac{6}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{12}{7}a+\left(\frac{6}{7}\right)^{2}=1+\left(\frac{6}{7}\right)^{2}$$

Square \frac{6}{7} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{12}{7}a+\frac{36}{49}=1+\frac{36}{49}$$

Add 1 to \frac{36}{49}.

$$a^{2}+\frac{12}{7}a+\frac{36}{49}=\frac{85}{49}$$

Factor a^{2}+\frac{12}{7}a+\frac{36}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+\frac{6}{7}\right)^{2}=\frac{85}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{6}{7}\right)^{2}}=\sqrt{\frac{85}{49}}$$

Simplify.

$$a+\frac{6}{7}=\frac{\sqrt{85}}{7} a+\frac{6}{7}=-\frac{\sqrt{85}}{7}$$

Subtract \frac{6}{7} from both sides of the equation.

$$a=\frac{\sqrt{85}-6}{7} a=\frac{-\sqrt{85}-6}{7}$$