1. Quadratic Equations
Definition:
A quadratic equation is an equation of the form:
ax2+bx+c=0ax2+bx+c=0
where aa, bb, and cc are constants and a≠0a =0.
Solving Quadratic Equations:
1. Factoring Method:
• Factor the quadratic expression ax2+bx+cax2+bx+c into two binomials.
• Set each factor equal to zero and solve for xx. Example: Solve
x2+5x+6=0x2+5x+6=0.
(x+2)(x+3)=0(x+2)(x+3)=0
x+2=0⇒x=−2x+2=0⇒x=−2 x+3=0⇒x=−3x+3=0⇒x=−3 Solution: x=−2x=−2 and
x=−3x=−3
2. Completing the Square:
• Rewrite the quadratic equation in the form (x+p)2=q(x+p)2=q.
• Take the square root of both sides and solve for xx. Example: Solve
x2+6x−7=0x2+6x−7=0.
x2+6x=7(Move -7 to the other side)x2+6x=7(Move -7 to the other side)(x+3)2=16(Add 9
to both sides to complete the square)(x+3)2=16(Add 9 to both sides to complete the
square)
Taking the square root:
x+3=±4x+3=±4
So, x=1x=1 or x=−7x=−7.
Definition:
A quadratic equation is an equation of the form:
ax2+bx+c=0ax2+bx+c=0
where aa, bb, and cc are constants and a≠0a =0.
Solving Quadratic Equations:
1. Factoring Method:
• Factor the quadratic expression ax2+bx+cax2+bx+c into two binomials.
• Set each factor equal to zero and solve for xx. Example: Solve
x2+5x+6=0x2+5x+6=0.
(x+2)(x+3)=0(x+2)(x+3)=0
x+2=0⇒x=−2x+2=0⇒x=−2 x+3=0⇒x=−3x+3=0⇒x=−3 Solution: x=−2x=−2 and
x=−3x=−3
2. Completing the Square:
• Rewrite the quadratic equation in the form (x+p)2=q(x+p)2=q.
• Take the square root of both sides and solve for xx. Example: Solve
x2+6x−7=0x2+6x−7=0.
x2+6x=7(Move -7 to the other side)x2+6x=7(Move -7 to the other side)(x+3)2=16(Add 9
to both sides to complete the square)(x+3)2=16(Add 9 to both sides to complete the
square)
Taking the square root:
x+3=±4x+3=±4
So, x=1x=1 or x=−7x=−7.
