Solve 8^2+(9a)^2=10^2

$$8 ^ { 2 } + ( 9 a ) ^ { 2 } = 10 ^ { 2 }$$

Solve for a

$$a=-\frac{2}{3}\approx -0.666666667$$

$$a=\frac{2}{3}\approx 0.666666667$$

Steps Using Difference of Squares

Calculate 8 to the power of 2 and get 64.

$$64+\left(9a\right)^{2}=10^{2}$$

Expand \left(9a\right)^{2}.

$$64+9^{2}a^{2}=10^{2}$$

Calculate 9 to the power of 2 and get 81.

$$64+81a^{2}=10^{2}$$

Calculate 10 to the power of 2 and get 100.

$$64+81a^{2}=100$$

Subtract 100 from both sides.

$$64+81a^{2}-100=0$$

Subtract 100 from 64 to get -36.

$$-36+81a^{2}=0$$

Divide both sides by 9.

$$-4+9a^{2}=0$$

Consider -4+9a^{2}. Rewrite -4+9a^{2} as \left(3a\right)^{2}-2^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(3a-2\right)\left(3a+2\right)=0$$

To find equation solutions, solve 3a-2=0 and 3a+2=0.

$$a=\frac{2}{3} a=-\frac{2}{3}$$

Steps by Finding Square Root

Calculate 8 to the power of 2 and get 64.

$$64+\left(9a\right)^{2}=10^{2}$$

Expand \left(9a\right)^{2}.

$$64+9^{2}a^{2}=10^{2}$$

Calculate 9 to the power of 2 and get 81.

$$64+81a^{2}=10^{2}$$

Calculate 10 to the power of 2 and get 100.

$$64+81a^{2}=100$$

Subtract 64 from both sides.

$$81a^{2}=100-64$$

Subtract 64 from 100 to get 36.

$$81a^{2}=36$$

Divide both sides by 81.

$$a^{2}=\frac{36}{81}$$

Reduce the fraction \frac{36}{81} to lowest terms by extracting and canceling out 9.

$$a^{2}=\frac{4}{9}$$

Take the square root of both sides of the equation.

$$a=\frac{2}{3} a=-\frac{2}{3}$$

Steps Using the Quadratic Formula

Calculate 8 to the power of 2 and get 64.

$$64+\left(9a\right)^{2}=10^{2}$$

Expand \left(9a\right)^{2}.

$$64+9^{2}a^{2}=10^{2}$$

Calculate 9 to the power of 2 and get 81.

$$64+81a^{2}=10^{2}$$

Calculate 10 to the power of 2 and get 100.

$$64+81a^{2}=100$$

Subtract 100 from both sides.

$$64+81a^{2}-100=0$$

Subtract 100 from 64 to get -36.

$$-36+81a^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$81a^{2}-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 81 for a, 0 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{0±\sqrt{0^{2}-4\times 81\left(-36\right)}}{2\times 81}$$

Square 0.

$$a=\frac{0±\sqrt{-4\times 81\left(-36\right)}}{2\times 81}$$

Multiply -4 times 81.

$$a=\frac{0±\sqrt{-324\left(-36\right)}}{2\times 81}$$

Multiply -324 times -36.

$$a=\frac{0±\sqrt{11664}}{2\times 81}$$

Take the square root of 11664.

$$a=\frac{0±108}{2\times 81}$$

Multiply 2 times 81.

$$a=\frac{0±108}{162}$$

Now solve the equation a=\frac{0±108}{162} when ± is plus. Reduce the fraction \frac{108}{162} to lowest terms by extracting and canceling out 54.

$$a=\frac{2}{3}$$

Now solve the equation a=\frac{0±108}{162} when ± is minus. Reduce the fraction \frac{-108}{162} to lowest terms by extracting and canceling out 54.

$$a=-\frac{2}{3}$$

The equation is now solved.

$$a=\frac{2}{3} a=-\frac{2}{3}$$