Download Quadratic Equations

Quadratic Formula
N-CN.7 Solve quadratic equations with real coefficients that have complex
solutions.
A-REI.4 Solve quadratic equations in one variable.
Use the method of completing the square to transform any quadratic equation
in x into an equation of the form (x – p)2 = q that has the same solutions. Derive
the quadratic formula from this form.
Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots,
completing the square, the quadratic formula and factoring, as appropriate to
the initial form of the equation. Recognize when the quadratic formula gives
complex solutions and write them as a ± bi for real numbers a and b.
Discriminant
• Tells the type and number of solutions to a
quadratic equation.
Discriminant
• Negative number: 2 complex solutions
• Zero: 1 real solution
• Positive number: 2 real solutions
• Perfect square: rational
• Non perfect square: irrational
Examples
EX: 5𝑥 2 − 𝑥 − 1 = 0
EX: 𝑥 2 + 3𝑥 = −6
EX: 2𝑥 2 − 𝑥 − 15 = 0
EX: 𝑥 2 = 16𝑥 − 64
Quadratic Formula
• Use to solve quadratic equations that CANNOT
be factored
• Set = 0
• Substitute values of a, b, and c into formula,
then simplify.
−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄
𝒙=
𝟐𝒂
Examples
EX: 𝑥 2 − 𝑥 + 1 = 0
EX: 3𝑥 2 + 8𝑥 = 3
EX: 𝑥 2 − 6𝑥 + 3 = 0
EX: 2𝑥 2 + 4 = 7𝑥
Word Problems
• An object is dropped from a height of 1700 feet
above the ground. The function ℎ = −16𝑡 2 +
1700 gives the object’s height h in feet during
free fall at t seconds. When will the object be
1000 feet above the ground?
• The quadratic function ℎ = −0.01𝑥 2 + 1.18𝑥 +
2 models the height of a punted football. The
horizontal distance in feet from the point of
impact with the kicker’s foot is x and h is the
height of the ball in feet. If the nearest defensive
player is 5 feet from the point of impact, how
high must the player reach to block the punt?
Solving Quadratics Review
• Is it missing the middle term?
– Solve by taking the square root
• Can you factor?
– GCF, Backwards FOIL, Difference of Squares,
Bottoms Up
• If you cannot use square root or factor:
– Solve by completing the square (only if no a)
– Solve using the quadratic formula
Examples
• Solve the quadratic equations by factoring, using square
roots, completing the square, or using the quadratic formula.
EX: 𝑥 2 + 49 = 0
EX: 𝑥 2 − 2𝑥 − 17 = 0
EX: 𝑥 2 − 𝑥 = 30
EX: 3𝑏2 = 10𝑏 + 32
EX: 2𝑥 2 + 2𝑥 + 3 = 0
EX: 3𝑥 2 + 36 = 0