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Math · 07 · Advanced Math · Free

Quadratic Equations

Notes

Key points

  • Standard form ax2+bx+c=0ax^2+bx+c=0: get everything on one side first.
  • Factor when the numbers are friendly; use the formula when they aren't.
  • Discriminant b2−4acb^2-4ac decides how many real solutions.
  • Sum of roots −ba-\tfrac{b}{a}, product ca\tfrac{c}{a} — no solving needed.
  • Vertex form a(x−h)2+ka(x-h)^2+k shows the min or max straight away.
  • Desmos: graph it, read the x-intercepts. Usually the fastest route.

Formulas

Quadratic formulax=−b±b2−4ac2ax=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}
DiscriminantD=b2−4acD=b^2-4acD > 0: two, D = 0: one, D < 0: none
Sum and product of rootsr1+r2=−bar1r2=car_1+r_2=-\dfrac{b}{a}\qquad r_1r_2=\dfrac{c}{a}
Vertexx=−b2ax=-\dfrac{b}{2a}
Difference of squaresa2−b2=(a+b)(a−b)a^2-b^2=(a+b)(a-b)
Perfect square(a±b)2=a2±2ab+b2(a\pm b)^2=a^2\pm 2ab+b^2

Common traps

  • Dividing both sides by xx — you lose the x=0x=0 solution.
  • Sign slips: −b-b flips the sign of bb, and b2b^2 is always positive.
  • Answering with one root when the question asks for the sum, product or positive one.

SAT tip

“How many solutions” or “no real solution” questions are discriminant questions — skip solving entirely.

Test yourself

10 official SAT questions on Quadratic Equations, easiest first.

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