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High SchoolAlgebra
Quadratic Equations
Equations with x² — factoring and the formula
Solve quadratic equations by factoring, completing the square and the quadratic formula.
✓ Standard form: ax² + bx + c = 0✓ Factoring: (x+p)(x+q) = 0 → x = −p or −q✓ Quadratic formula: x = (−b ± √(b²−4ac)) ...
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📖 Understanding Quadratic Equations
A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0. The highest power of x is 2. Solutions are called roots.
Method 1 — Factoring: Rewrite as (x + p)(x + q) = 0. Then x = −p or x = −q. Works when the equation factors neatly.
Method 2 — Quadratic Formula: x = (−b ± √(b²−4ac)) / 2a. Always works! The discriminant b²−4ac tells us: if positive → 2 real roots, if zero → 1 root, if negative → no real roots.
Method 3 — Completing the Square: Rewrite as (x + p)² = q, then solve. Also used to find the vertex of a parabola.
🔑 Key Points to Remember
- ✓Standard form: ax² + bx + c = 0
- ✓Factoring: (x+p)(x+q) = 0 → x = −p or −q
- ✓Quadratic formula: x = (−b ± √(b²−4ac)) / 2a
- ✓Discriminant (b²−4ac): >0 two roots, =0 one root, <0 no real roots
- ✓Parabola opens up if a>0, down if a<0
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