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How to Solve a Quadratic Equation
A quadratic equation has the form ax² + bx + c = 0 and is the second-most-common equation type after linear. Three methods solve every one: factoring (fastest when it works), the quadratic formula (always works), and completing the square (proves the formula and helps with graphing).
When the method applies
- • Equation has an x² term you cannot eliminate
- • Factoring does not seem to work
- • Discriminant looks ugly (not a perfect square)
- • Word problem reduces to ax² + bx + c = 0
Common mistakes
- • Quadratic does not factor over the integers
- • Coefficients are messy fractions or decimals
- • Student forgot the ± in the quadratic formula
- • Wrong sign inside the square root
Step-by-step method
- • Always try factoring first if a=1 and constant has small factor pairs
- • If factoring fails, use the quadratic formula: x = (-b ± √(b²-4ac)) / 2a
- • Completing the square works for any quadratic and proves the formula
- • Check answers by plugging back into the original equation
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Build long-term fluency
- • Memorise the quadratic formula cold (sing the song)
- • Practice 10 factoring drills weekly until pattern recognition is fast
- • Always compute the discriminant first to predict # of real roots
Edge cases & deeper reading
If the discriminant is negative you have complex roots — see Algebra II or check whether the original problem allows them. If coefficients are symbolic (letters), the formula still applies — just substitute carefully.
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