Determine a rotation $\mathbf{x}=P \mathbf{x}^{\prime}$ that removes the cross-product terms, and give its equation in the $x^{\prime} y^{\prime} z^{\prime}$-system. Name the quadric.

$$2 x^2+3 y^2+23 z^2+72 x z+150=0$$

Using the procedure in the previous exercise to solve $y^{\prime \prime}+y^{\prime}-12 y=0$.

The owner of a rapidly expanding business finds that for the first five months of the year the sales (in thousands) are $3.0,$3.5, $5.0,$6.2, and $7.5. The owner plots these figures on a graph and conjectures that for the rest of the year, the sales curve can be approximated by a quadratic polynomial. Find the least squares quadratic polynomial fit to the sales curve, and use it to project the sales for the twelfth month of the year.

Using the result in the previous exercise show that

$$1^2+2^2+3^2+\cdots+(n-1)^2=\frac{(n-1) n(2 n-1)}{6}$$

Using Theorem 9.5.3 find which of the matrices in the previous question are positive definite.

Find the translation equations that will put the quadric in standard position, and find the equation of the quadric in the translated coordinate system. Name the quadric. $x^2+y^2+z^2-2 x+4 y-6 z=11$

A translation will put the conic in standard position. give the name of the conic and give its equation in the translated coordinate system.

$$x^2+10 x+7 y=-32$$

Find the translation equations that will put the quadric in standard position, and find the equation of the quadric in the translated coordinate system. Name the quadric.

$$7 x^2-3 y^2+126 x+72 y+z+135=0$$

A translation will put the conic in standard position. give the name of the conic and give its equation in the translated coordinate system.

$$2 x^2-3 y^2+6 x+20 y=-41$$

Find the translation equations that will put the quadric in standard position, and find the equation of the quadric in the translated coordinate system. Name the quadric.

$$x^2+16 y^2+2 x-32 y-16 z-15=0$$

Express the formula

$$1^2+2^2+3^2+\cdots+n^2=\frac{n(n+1)(2 n+1)}{6}$$

using the following steps. Solve the equation in (c) for $1^2+2^2+3^2+\cdots+n^2$, use the result of a previous exercise, and then simplify.

A translation will put the conic in standard position. give the name of the conic and give its equation in the translated coordinate system.

$$x^2+y^2+6 x-10 y+18=0$$

Find the translation equations that will put the quadric in standard position, and find the equation of the quadric in the translated coordinate system. Name the quadric.

$$4 x^2+9 y^2-z^2-54 y-50 z=544$$

Express the formula

$$1^2+2^2+3^2+\cdots+n^2=\frac{n(n+1)(2 n+1)}{6}$$

using the following steps. Apply (a) to each term on the left side of (b) to show that

$$(n+1)^3-1=3\left[1^2+2^2+3^2+\cdots+n^2\right]+3[1+2+3+\cdots+n]+n$$