grade 9 Solving Quadratic Equations

Solving Quadratic Equations

A quadratic equation has highest power 2 in x, written ax² + bx + c = 0. Finding when a thrown object reaches height zero usually gives two answers. Factor so a product of two linear expressions equals zero, or, when that fails, use the quadratic formula. Keep a at 1 and change only b and c: defaults b=−2, c=−3 give D=16 and roots 3 and −1.

Roots = Where the Parabola Meets the x-axis

b (linear coeff)-2

c (constant)-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

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'

⑤Sum of roots=−b/a, product=c/a. Integer roots need D a perfect square (default D=16=4²)

Work It Out

Example 1

Solve x²−5x+6=0.

Factor the left side: product 6, sum −5 gives −2 and −3.

x²−5x+6 = (x−2)(x−3)

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.

Substitute a=1, b=−4, c=1.

x = (4±√(16−4))/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?

①3

②6

③9

④12

⑤36

▸ ③ 9

A double root means the discriminant D=0.

D = (−6)²−4·1·k = 36−4k

Solve 36−4k=0.

36−4k=0 ⇒ k=9

🎯 Exam Points

①Try factoring first; otherwise use the formula

②Double root iff D = 0

③Vieta: sum of roots = −b/a, product = c/a

④Integer roots ⇔ D is a perfect square

⑤Always start with the right side equal to 0

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Factorization

Applications of Quadratic Equations

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