Solve quadratic equations using factorisation and the quadratic formula, and understand the nature of roots.
A quadratic equation ax²+bx+c=0 can be solved by factorising into two linear factors, if possible.
Solve: x²−5x+6=0
(x−2)(x−3)=0; x=2 or x=3
Solve: x²+3x−10=0
(x+5)(x−2)=0; x=−5 or x=2
x=[−b±√(b²−4ac)]/2a solves any quadratic equation. The discriminant b²−4ac tells us the nature of the roots.
Solve x²+2x−8=0 using the quadratic formula.
x=[−2±√(4+32)]/2=[−2±6]/2; x=2 or x=−4
Find the discriminant of 2x²−4x+2=0 and describe the roots.
D=16−16=0; equal (repeated) real roots
1. Solve by factorisation: x²−7x+12=0
2. Solve by factorisation: x²−x−6=0
3. Solve using the formula: x²−4x+1=0
4. Find the discriminant of x²+6x+9=0 and describe the roots.
5. Find the discriminant of x²+x+1=0 and describe the roots.