Every Math Formula You Need to Memorize for the SAT

Main Ideas

• The video is a checklist of SAT Math formulas the presenter says are worth knowing, rather than a full lesson on how to solve every type of problem.
• Many geometry formulas—including the Pythagorean theorem, special right triangles, and volume formulas—are provided on the SAT reference sheet. The presenter recommends becoming comfortable using them but says they do not need to be memorized.
• Desmos can handle or check some calculations, so the presenter argues that fewer formulas need to be memorized than on the paper SAT. Still, Desmos should not replace understanding what the formulas mean.
• The presenter describes some topics as especially useful and others—such as the quadratic formula, completing the square, and the discriminant—as rare on the digital SAT. The sum and product of a quadratic’s roots are highlighted as surprisingly useful.

Formulas and Concepts Covered

Linear Equations

• Slope-intercept form: (y=mx+b)
  • (m) is the slope; (b) is the (y)-intercept.
• Standard form: (Ax+By=C)
  • When (B\ne 0), the slope is (-A/B) and the (y)-intercept is (C/B).

Exponential Functions and Exponent Rules

• Exponential form: Commonly written as (y=a\cdot b^x), where (a) gives the value at (x=0), and (b) determines whether the function grows or decays.
• Rules for powers with the same base:
  • Multiply: (x^m x^n=x^{m+n})
  • Divide: (x^m/x^n=x^{m-n})
  • Raise a power to a power: ((x^m)^n=x^{mn})
• Additional rules:
  • (x^0=1) for nonzero (x)
  • (x^{-1}=1/x)
  • (x^{1/2}=\sqrt{x}) and (x^{1/3}=\sqrt[3]{x})

The presenter demonstrates rewriting roots as fractional exponents and combining exponents to solve for an unknown exponent.

Quadratics

• Standard form: (y=ax^2+bx+c)
  • (c) is the (y)-intercept.
  • The (x)-coordinate of the vertex is (-b/(2a)).

• Quadratic formula: [ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} ] It gives the equation’s solutions (the (x)-intercepts). The presenter says it is rarely unavoidable on the digital SAT but is worth recognizing.

• Completing the square: [ x^2+bx+c=\left(x+\frac b2\right)^2+c-\left(\frac b2\right)^2 ] This can be used to rewrite a quadratic, including to put a circle equation into standard form.

• Vertex form: (y=a(x-h)^2+k), with vertex ((h,k)).

• Discriminant: (b^2-4ac)
  • Positive: two real solutions.
  • Zero: one real solution.
  • Negative: no real solutions.

The presenter says the discriminant is rarely needed and that Desmos may help determine the number of solutions.

• Factored form: (y=a(x-p)(x-q)), where (p) and (q) are the roots.
  • Sum of roots: (-b/a)
  • Product of roots: (c/a)

The video uses these relationships to set up an equation involving a quadratic’s roots.

Circles, Averages, and Percent Change

• Circle: ((x-h)^2+(y-k)^2=r^2)
  • Center: ((h,k))
  • Radius: (r)
• Arithmetic mean: The sum of the values divided by the number of values.
• Percent change: [ \frac{\text{new value}-\text{old value}}{\text{old value}}\times100\% ]

The example compares 35,000 with 43,000, giving an increase of about 23%.

Geometry and Trigonometry

• Exterior angle theorem: A triangle’s exterior angle equals the sum of its two non-adjacent interior angles.
• Angle bisector theorem: If an angle bisector meets the opposite side, it divides that side in the ratio of the two sides forming the bisected angle. The video illustrates combining this proportion with a perimeter equation to find a side length.
• Interior angle sum of a polygon: ((n-2)\times180^\circ), where (n) is the number of sides.
• Surface area:
  • Sphere: (4\pi r^2)
  • Cone: (\pi r^2+\pi r\ell), where (\ell) is the slant height. For a right cone, (\ell=\sqrt{h^2+r^2}).

The presenter says the SAT reference sheet does not provide surface-area formulas and recommends memorizing the sphere and cone formulas, while being able to work out surface areas of shapes such as cubes and pyramids.

• SOHCAHTOA for a right triangle:
  • (\sin\theta=\text{opposite}/\text{hypotenuse})
  • (\cos\theta=\text{adjacent}/\text{hypotenuse})
  • (\tan\theta=\text{opposite}/\text{adjacent})
• Complementary-angle identity: (\sin X=\cos(90^\circ-X))

The presenter demonstrates using this relationship together with another equation to solve for two acute angles.

Speakers and Sources

• Speaker: One unnamed presenter, speaking as an SAT Math tutor and host of the channel.
• Sources and tools mentioned: The SAT formula/reference sheet and Desmos. No additional speakers are featured.