A quadratic equation is one whose highest power of x is 2, written as ax² + bx + c = 0.
Like finding when a thrown object reaches height zero, it usually has two answers.
Factor it so a product of two linear expressions equals zero, or, when nothing factors, use the quadratic formula.
Change the coefficients to see how the two roots shift and how the discriminant sets their number.
Roots = Where the Parabola Meets the x-axis
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Where a parabola crosses the x-axis are the roots
-2
-3
👀 Roots from the Graph
①Parabola meets x-axis twice → two distinct real roots
②Touches once → one repeated root
③Misses entirely → no real roots
Solving by Factoring
Factoring Method
ax²+bx+c = 0 → a(x−p)(x−q) = 0 → x=p or x=q
Factor and set each factor to zero
🧩 Factoring Steps
①Move everything to one side (= 0)
②Factor the left
③Set each factor = 0
④e.g., x²−5x+6 = 0 → (x−2)(x−3) = 0 → x = 2, 3
Completing the Square & Square Roots
Square Root Method
(x+a)² = k → x+a = ±√k → x = −a ± √k
Convert to perfect-square form, then take roots
💡 How to Complete the Square
①x²+6x = 7 → x²+6x+9 = 16 (add 9 to both sides)
②(x+3)² = 16
③x+3 = ±4
④x = 1 or x = −7
Quadratic Formula
Quadratic Formula
x = −b ± √(b²−4ac)2a
Universal formula that always works
Discriminant
D = b²−4ac
D>0: two real roots; D=0: double root; D<0: no real roots
⚠️ Using the Formula Carefully
①Move to ax²+bx+c = 0 first
②Watch the −b sign
③Divide everything by 2a (not just by 2)
④If D<0, write 'no real roots'
Work It Out
Example 1
Solve x²−5x+6=0.
1
Factor the left side: product 6, sum −5 gives −2 and −3.
x²−5x+6 = (x−2)(x−3)
2
Set each factor to zero.
(x−2)(x−3)=0 ⇒ x=2 or x=3
▸ x = 2 or x = 3
When it factors, this is the fastest route.
Example 2
Solve x²−4x+1=0 with the quadratic formula.
1
Substitute a=1, b=−4, c=1.
x = (4±√(16−4))/2
2
Since √12=2√3, simplify.
x = (4±2√3)/2 = 2±√3
▸ x = 2 ± √3
If it does not factor, use the formula; simplify under the root first.
Exam Wrap-up
Three Solving Methods
Factor → Complete the Square → Quadratic Formula
If factoring fails, use the formula
G9 school-exam type
If x²−6x+k=0 has a double root, what is the constant k?