high school Various Equations

Various Equations

Beyond quadratics lie cubic and quartic equations and systems. A cubic's roots are where its graph meets the x-axis, so you find one by the Factor Theorem and reduce to a quadratic by synthetic division. A system's solutions are where two graphs meet. Absolute-value and radical equations continue in the formula boxes below. Change the coefficients and the cubic's real roots and the two curves appear.

Cubic Equations — Roots from the Graph

The roots of x³ + px + q = 0 are where the graph meets the x-axis. Watch how the root count changes as the coefficients change.

p (x coefficient)-3

q (constant)2

💡 Cubic Roots

①A cubic always has at least one real root

②With strong curvature (large |p|), three real roots are possible

③Use the Factor Theorem to find one root, then synthetic division to reduce to a quadratic

Solving Cubics & Quartics

Factoring Strategy

integer root candidates → synthetic division → reduce to quadratic

Plug in divisors of the constant term

Substitution

x⁴ + ax² + b = 0 → substitute t = x²

Quartic becomes a quadratic

🔑 Solving Order

①Check common factors

②Try integer root candidates (divisors of the constant)

③Synthetic division to lower degree

④Apply the quadratic formula to whatever remains

Systems — Intersections of Curves

A system's solutions are the intersections of the graphs. Linear-quadratic systems can be solved by substitution.

slope k2

📐 Solving Systems

①Substitute one equation into the other to get a single-variable equation

②x² = kx + 1 → x² − kx − 1 = 0

③This sketch is y = x² and y = kx + 1, so D = k² + 4 is always positive and there are 2 intersections

④Intersection coordinates = solutions

Special Equations

Absolute Value Equations

|f(x)| = g(x) → f(x) = ±g(x)

Split into two cases (with g(x) ≥ 0)

Radical Equations

√(f(x)) = g(x) → f(x) = g(x)², g(x) ≥ 0

Always verify after squaring

Work It Out

Example 1

Find the real root of the cubic x³ − 8 = 0.

Factor using the difference of cubes.

x³ − 2³ = (x − 2)(x² + 2x + 4) = 0

x − 2 = 0 gives the real root; x² + 2x + 4 = 0 has D = 4 − 16 < 0, so complex roots.

x = 2 (x² + 2x + 4 = 0 has complex roots)

▸ The real root is x = 2

For a cubic, factor first, then use the discriminant to separate real and complex roots.

Example 2

Solve the system x + y = 5, x² + y² = 13.

From x + y = 5, substitute y = 5 − x into the second equation.

x² + (5 − x)² = 13

This simplifies to x² − 5x + 6 = 0, i.e. (x − 2)(x − 3) = 0.

x² − 5x + 6 = 0 ⇒ (x, y) = (2, 3) or (3, 2)

▸ (x, y) = (2, 3) or (3, 2)

Solve the linear equation for one variable and substitute into the quadratic.

Wrap-up

High-degree Solving Core

Factor Theorem + synthetic division → reduce the degree

Break cubics/quartics into quadratics

Grade-10 education-office assessment type

Find the sum of the three real roots of x³ − 6x² + 11x − 6 = 0.

①3

②4

③5

④6

⑤7

▸ ④ 6

For ax³ + bx² + cx + d = 0, the sum of the roots is −b/a.

α + β + γ = −b/a

Substitute a = 1, b = −6.

α + β + γ = −(−6)/1 = 6

🎯 Exam Points

①Cubic: always at least one real root

②Quartic: try t = x² substitution (when only even powers)

③Systems: substitution to single variable

④Radical: square both sides, then verify (extraneous roots!)

⑤Absolute value: case-split, then check each condition

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Quadratic Equations

Various Inequalities

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