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Grade 10 / High 1 (age 15-16)

high school Quadratic Equations

Quadratic Equations

A quadratic's discriminant D = b² − 4ac reveals how many roots it has before you solve it. Positive D gives two real roots, zero a repeated root, negative complex roots, seen in how the parabola meets the x-axis. Vieta's relations then give the sum and product of the roots from the coefficients alone. Drag the coefficients and the vertex and the parabola and its roots respond.

Discriminant — Counts the Roots

For ax² + bx + c = 0, the discriminant D = b² − 4ac tells you the existence of real roots. Visualize how the parabola meets the x-axis.

-2
-3
💡 Meaning of D
①D > 0: parabola meets x-axis at 2 points → 2 distinct real roots
②D = 0: parabola is tangent → repeated root
③D < 0: no intersection → complex roots
④In this sketch a is 1, so D = b² − 4c

Quadratic Formula

Quadratic Formula
x = -b ± √(b² - 4ac)2a
Solutions of ax² + bx + c = 0
Discriminant
D = b² − 4ac
D > 0: two real, D = 0: repeated, D < 0: complex
📐 Deriving the Formula
①Complete the square: (x + b/2a)² = (b²−4ac)/4a²
②Take the square root: x + b/2a = ±√D / 2a
③Solve for x

Vieta — Roots & Coefficients

Without solving for α, β, you can read off the sum and product from the coefficients.

-5
6

Vieta Formulas

Sum & Product of Roots
α + β = -ba, αβ = ca
For ax² + bx + c = 0
Rebuild from Roots
x² − (α+β)x + αβ = 0
Sum and product determine the quadratic
🔑 Useful Identities
①α² + β² = (α+β)² − 2αβ
②1/α + 1/β = (α+β) / αβ
③|α − β| = √((α+β)² − 4αβ) = √D / |a|

Work It Out

Example 1
Find the two roots of the quadratic x² − 5x + 6 = 0.
1
Factor the left side.
x² − 5x + 6 = (x − 2)(x − 3) = 0
2
Set each factor to 0.
x = 2 or x = 3
▸ x = 2 or x = 3
When it factors, that is faster than the quadratic formula.
Example 2
Let α, β be the roots of x² − 7x + 10 = 0. Find α² + β².
1
Get the sum and product from the relations between roots and coefficients.
α + β = 7, αβ = 10
2
Substitute into α² + β² = (α + β)² − 2αβ.
= 7² − 2·10 = 49 − 20 = 29
▸ α² + β² = 29
Compute the symmetric expression from sum and product, without finding the roots.

Wrap-up

Quadratic Formula
x = -b ± √(b² - 4ac)2a
Universal solver for quadratics
Vieta
α + β = -ba, αβ = ca
Sum/product without finding roots
Grade-10 education-office assessment type
If x² − 4x + k = 0 has a double root, what is the constant k?
①1
②2
③3
④4
⑤5
▸ ④ 4
1
A double root means the discriminant D = 0.
D = b² − 4ac = (−4)² − 4·1·k
2
Solve D = 0 for k.
16 − 4k = 0 ⇒ k = 4
🎯 Exam Points
①Use D first to decide whether real roots exist
②D = 0 leads to tangency / repeated-root problems
③Use Vieta for symmetric expressions in α, β
④Sum & product → write down the new quadratic
⑤Complex roots come in conjugate pairs
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Complex Numbers
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Various Equations
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