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If the roots of the equation x2−5x+6=0 are α and β, find the value of α2+β2:
13
6
12
5
5
Since the equation factors easily into (x-2)(x-3)=0, the roots are 2 and 3. Simply calculate 2² + 3² = 4 + 9 = 13.
A quadratic equation x² - 5x + 6 = 0 with roots alpha and beta.
α+β=−ab,αβ=ac,α2+β2=(α+β)2−2αβ
Since the equation factors easily into (x-2)(x-3)=0, the roots are 2 and 3. Simply calculate 2² + 3² = 4 + 9 = 13.
Students often miscalculate by finding (alpha + beta)2 = 25 and subtracting alphabeta = 6 instead of 2alpha*beta = 12.
Identify coefficients
For the quadratic equation ax2+bx+c=0, here a=1, b=−5, and c=6.
a=1,b=−5,c=6
Apply Vieta's formulas
The sum of roots is α+β=−b/a=5 and the product of roots is αβ=c/a=6.
α+β=5,αβ=6
Calculate sum of squares
Use the identity α2+β2=(α+β)2−2αβ to substitute the values: 52−2(6)=25−12.
25−12=13
A is correct because the calculated value of 13 matches option A.
This identity (α+β)2−2αβ is frequently used in coordinate geometry to find the length of chords or distances between roots.
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