You can be
great at math.

The minus sign belongs to the $1$, not to the whole sum. Because $-1$ and $8$ have opposite signs, they partly cancel rather than combining into a larger negative number.

Think of a number line: start at $-1$. Adding positive $8$ means moving 8 units to the right. Where do you land?

The first error is in the expansion. The $3$ multiplies both terms inside the parentheses, including $-4$:

So your equation becomes

Your later algebra was consistent with the expansion you wrote; it’s that expansion that changed the equation. What do you get if you subtract $2x$ from both sides of the corrected equation?

Not quite: your setup uses $125^\circ$ as though it were an interior angle, but it’s an exterior angle.

An exterior angle equals the sum of the two interior angles that aren’t next to it. So if $x$ is the missing angle, we have

What operation would you use to isolate $x$?

You mixed up an exponent with multiplication. A logarithm tells us what exponent to put on the base.

means “8 raised to what power equals $x$?” The logarithm says that power is 4, so the correct rewrite is

That means multiplying four factors of 8, rather than multiplying 8 by 4. Can you evaluate $8^4$?

Your multiplication is correct, but the ratio is being used in the wrong direction. Cosine means

So $\cos C=3/5$ tells us the adjacent side is $3/5$ of the hypotenuse, not the other way around. With adjacent side $30$, we have

or equivalently,

Also, $18$ can’t be the hypotenuse because the hypotenuse must be longer than the adjacent side, $30$.

What operation would undo multiplying $h$ by $3/5$?

You’re right that $y=0$, because the angle $\pi$ points along the x-axis. The key is which direction along that axis.

In polar coordinates, $r=4$ means move 4 units in the direction given by the angle. Since $\pi$ radians is $180^\circ$, that direction is left, along the negative x-axis. So $r=4$ doesn’t mean $x=4$.

Using the conversion formulas:

The rectangular point is therefore $\boxed{(-4,0)}$: 4 units left of the origin.

The key is how many standard deviations the endpoints are from the mean, not how many intervals are written.

Both endpoints are $56$ units from $150$. Since the standard deviation is $28$,

So $94$ to $206$ covers two standard deviations below and above the mean. The $68\%$ rule applies within one standard deviation of the mean.

According to the empirical rule, what percentage falls within two standard deviations?

Not quite. Canceling works for factors multiplying the whole numerator and denominator. In $x-6$, the $x$ is only one term, so we can’t cancel it out of the whole numerator.

Instead, divide each numerator term by $x$:

Since the denominator isn’t zero at $x=3$, we can find the limit by substitution. What do you get when you substitute $3$ for $x$ in this corrected expression?

Your choice is valid, but differentiating $\cos x$ keeps producing trig functions, while integrating $x$ raises its degree. That makes the remaining integral harder.

For integration by parts, aim to choose $u$ so it gets simpler when differentiated, and $dv$ so it’s easy to integrate. Here, a better choice is

Then

Using the integration-by-parts formula,

we get

Now the remaining integral is simpler because the factor $x$ is gone. What antiderivative would you use for $\sin x$?

When taking a partial derivative with respect to $y$, we treat $x$ as a constant. So $4x^3$ stays unchanged, and only $y^2$ is differentiated:

Your derivative of $y^2$, which is $2y$, is correct. The error is changing $4x^3$ into $12x^2$: that would be differentiating it with respect to $x$, but here we’re differentiating with respect to $y$.

Not quite. Since $\tan^2 y$ is in the denominator on the right, we multiply both sides by $\tan^2 y$ to cancel it, then multiply by $dx$:

Now all the $y$-terms are on the left and all the $x$-terms are on the right, so the variables are separated.

Can you rewrite $\tan^2 y$ using the identity $1+\tan^2 y=\sec^2 y$ to prepare the left side for integration?

The symbol $R_1 \leftrightarrow R_2$ means swap the entire first and second rows, not just their first entries. Every entry moves with its row, including the number after the divider in an augmented matrix.

For example,

Try your swap again, moving each whole row into the other row’s position.