Solve (3/4a^{2}+5/2b^{2}-1/8c^{2})+(1/2b^{2}-frac{5}{4}c^{2}+a^2)+(frac{3}{8}c^2-frac{7}{4}b^2-a^2)

$$( \frac { 3 } { 4 } a ^ { 2 } + \frac { 5 } { 2 } b ^ { 2 } - \frac { 1 } { 8 } c ^ { 2 } ) + ( \frac { 1 } { 2 } b ^ { 2 } - \frac { 5 } { 4 } c ^ { 2 } + a ^ { 2 } ) + ( \frac { 3 } { 8 } c ^ { 2 } - \frac { 7 } { 4 } b ^ { 2 } - a ^ { 2 } )$$

Evaluate

$$\frac{3a^{2}}{4}+\frac{5b^{2}}{4}-c^{2}$$

Solution Steps

Combine \frac{5}{2}b^{2} and \frac{1}{2}b^{2} to get 3b^{2}.

$$\frac{3}{4}a^{2}+3b^{2}-\frac{1}{8}c^{2}-\frac{5}{4}c^{2}+a^{2}+\frac{3}{8}c^{2}-\frac{7}{4}b^{2}-a^{2}$$

Combine -\frac{1}{8}c^{2} and -\frac{5}{4}c^{2} to get -\frac{11}{8}c^{2}.

$$\frac{3}{4}a^{2}+3b^{2}-\frac{11}{8}c^{2}+a^{2}+\frac{3}{8}c^{2}-\frac{7}{4}b^{2}-a^{2}$$

Combine \frac{3}{4}a^{2} and a^{2} to get \frac{7}{4}a^{2}.

$$\frac{7}{4}a^{2}+3b^{2}-\frac{11}{8}c^{2}+\frac{3}{8}c^{2}-\frac{7}{4}b^{2}-a^{2}$$

Combine -\frac{11}{8}c^{2} and \frac{3}{8}c^{2} to get -c^{2}.

$$\frac{7}{4}a^{2}+3b^{2}-c^{2}-\frac{7}{4}b^{2}-a^{2}$$

Combine 3b^{2} and -\frac{7}{4}b^{2} to get \frac{5}{4}b^{2}.

$$\frac{7}{4}a^{2}+\frac{5}{4}b^{2}-c^{2}-a^{2}$$

Combine \frac{7}{4}a^{2} and -a^{2} to get \frac{3}{4}a^{2}.

$$\frac{3}{4}a^{2}+\frac{5}{4}b^{2}-c^{2}$$

Factor

$$\frac{3a^{2}+5b^{2}-4c^{2}}{4}$$

Solution Steps

Factor out \frac{1}{8}.

$$\frac{6a^{2}+20b^{2}-c^{2}+4b^{2}-10c^{2}+8a^{2}+3c^{2}-14b^{2}-8a^{2}}{8}$$

Consider 6a^{2}+20b^{2}-c^{2}+4b^{2}-10c^{2}+8a^{2}+3c^{2}-14b^{2}-8a^{2}. Multiply and combine like terms.

$$6a^{2}+10b^{2}-8c^{2}$$

Consider 6a^{2}+10b^{2}-8c^{2}. Factor out 2.

$$2\left(3a^{2}+5b^{2}-4c^{2}\right)$$

Rewrite the complete factored expression.

$$\frac{3a^{2}+5b^{2}-4c^{2}}{4}$$