Solve (-3+2sqrt{7})*(7/2-2sqrt{7})=

$$(-3+2 \sqrt{ 7 } ) \times ( \frac{ 7 }{ 2 } -2 \sqrt{ 7 } )=$$

Evaluate

$$13\sqrt{7}-\frac{77}{2}\approx -4.105232956$$

Solution Steps

Apply the distributive property by multiplying each term of -3+2\sqrt{7} by each term of \frac{7}{2}-2\sqrt{7}.

$$-3\times \frac{7}{2}+6\sqrt{7}+2\sqrt{7}\times \frac{7}{2}-4\left(\sqrt{7}\right)^{2}$$

Express -3\times \frac{7}{2} as a single fraction.

$$\frac{-3\times 7}{2}+6\sqrt{7}+2\sqrt{7}\times \frac{7}{2}-4\left(\sqrt{7}\right)^{2}$$

Multiply -3 and 7 to get -21.

$$\frac{-21}{2}+6\sqrt{7}+2\sqrt{7}\times \frac{7}{2}-4\left(\sqrt{7}\right)^{2}$$

Fraction \frac{-21}{2} can be rewritten as -\frac{21}{2} by extracting the negative sign.

$$-\frac{21}{2}+6\sqrt{7}+2\sqrt{7}\times \frac{7}{2}-4\left(\sqrt{7}\right)^{2}$$

Cancel out 2 and 2.

$$-\frac{21}{2}+6\sqrt{7}+7\sqrt{7}-4\left(\sqrt{7}\right)^{2}$$

Combine 6\sqrt{7} and 7\sqrt{7} to get 13\sqrt{7}.

$$-\frac{21}{2}+13\sqrt{7}-4\left(\sqrt{7}\right)^{2}$$

The square of \sqrt{7} is 7.

$$-\frac{21}{2}+13\sqrt{7}-4\times 7$$

Multiply -4 and 7 to get -28.

$$-\frac{21}{2}+13\sqrt{7}-28$$

Convert 28 to fraction \frac{56}{2}.

$$-\frac{21}{2}+13\sqrt{7}-\frac{56}{2}$$

Since -\frac{21}{2} and \frac{56}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-21-56}{2}+13\sqrt{7}$$

Subtract 56 from -21 to get -77.

$$-\frac{77}{2}+13\sqrt{7}$$

Factor

$$\frac{26 \sqrt{7} - 77}{2} = -4.105232956160322$$