Transcript Quadratic Equations

Quadratic Equations
A-REI.4 Solve quadratic equations in one variable.
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.
Vocabulary
• Quadratic Equation: any equation in the form
• Solution to a quadratic equation: x-intercepts, zeros, roots (where y = 0)
Solving by Factoring
Steps:
1. Move everything to the side that has 𝑥 2 .
2. Determine the best method of factoring (GCF, Backwards
FOIL, Bottoms Up, Perfect Square Trinomial or Difference of
Squares).
3. Factor completely.
4. Set each factor = 0 and solve.
GCF (Greatest Common Factor)
• All terms are divisible by the same number or variable
• ALWAYS check for GCF first in every problem
Steps:
• Find the GCF
• Divide all terms by the GCF
• Write the GCF in front and remaining terms in parentheses
EX: 25𝑥 2 − 10𝑥 = 0
Basic Factoring (Backwards FOIL)
• Three terms, no leading coefficient
Steps:
• Figure out the factors that multiply to give the c and add to
give the b
EX: 𝑥 2 − 13𝑥 + 30 = 0
EX: 2𝑥 2 − 14𝑥 + 12 = 0
Bottoms Up (Leading Coefficient)
• Three terms with a number in
front
Steps:
• Multiply a and c
• Replace c with new product
and drop a
• Factor using backwards FOIL
• Divide both factors by a
• Bottoms up
EX: 6𝑥 2 − 𝑥 − 2 = 0
EX: 4𝑥 2 − 23𝑥 + 15 = 0
Perfect Square Trinomial
• Three terms, first and last terms are perfect squares
Steps:
• Take square root of first and last terms and use the sign from
the middle term
EX: 4𝑥 2 + 20𝑥 + 25 = 0
Difference of Squares
• Two terms and both terms are perfect squares, minus sign in
the middle
Steps:
• Take square root of both terms
• One factor plus, one factor minus
EX: 16𝑥 2 − 1 = 0
Review
• Identify the correct method of factoring and solve.
1. 𝑥 2 = 18𝑥 − 81
5. 𝑥 2 − 169 = 0
2. 2𝑥 2 + 23𝑥 = −45
6. −30𝑥 − 48 = 3𝑥 2
3. 𝑥 2 − 25𝑥 = 0
7. 4𝑥 2 = 100
4. −𝑥 2 + 42 = −11𝑥
8. 6𝑥 2 − 44𝑥 + 48 = 0