Math Foundations: 18 flashcards
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1. What is the definition of a **rational number**?
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A number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
2. State the **Pythagorean theorem** for a right triangle with legs $a$, $b$ and hypotenuse $c$.
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$$a^2 + b^2 = c^2$$
3. Simplify: $(3x^2y)(-2xy^3)$
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$-6x^3y^4$
4. What is the **distance formula** between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane?
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$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$
5. Solve for $x$: $2x + 5 = 3x - 7$
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$x = 12$
6. What is the **slope** of the line passing through $(2, 3)$ and $(4, 7)$?
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$m = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2$
7. Factor completely: $x^2 - 9$
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$(x - 3)(x + 3)$
8. What is the **midpoint** of the segment joining $(1, 2)$ and $(5, 8)$?
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$(3, 5)$
9. State the **quadratic formula** for solving $ax^2 + bx + c = 0$.
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$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
10. Find the value of $\sqrt{144} + \sqrt{25}$.
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$12 + 5 = 17$
11. What is the **area** of a triangle with base $b$ and height $h$?
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$$A = \frac{1}{2}bh$$
12. Simplify: $\frac{x^5}{x^2}$ (for $x \neq 0$)
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$x^3$
13. If $f(x) = 2x^2 - 3x + 1$, find $f(2)$.
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$f(2) = 2(4) - 3(2) + 1 = 8 - 6 + 1 = 3$
14. What is the **volume** of a cylinder with radius $r$ and height $h$?
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$$V = \pi r^2 h$$
15. Solve the inequality: $3x - 4 < 8$
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$x < 4$
16. What is the **sum** of the interior angles of a polygon with $n$ sides?
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$$(n - 2) \times 180^\circ$$
17. Rationalize the denominator: $\frac{1}{\sqrt{5}}$
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$\frac{\sqrt{5}}{5}$
18. What is the **equation** of a line in slope-intercept form?
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$$y = mx + b$$ where $m$ is slope and $b$ is the $y$-intercept.
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