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Find the quadratic equation whose roots are 3 and -5.
x² + 2x - 15 = 0
x² - 2x - 15 = 0
x² - 2x + 15 = 0
x² + 2x + 15 = 0
x² + 2x - 15 = 0
Find the sum of roots (3+(−5)=−2) and product of roots (3×(−5)=−15), then substitute into x2−(sum)x+(product)=0 to instantly get x2+2x−15=0.
Roots of the quadratic equation are α=3and β=−5
x2−(α+β)x+αβ=0
Find the sum of roots (3+(−5)=−2) and product of roots (3×(−5)=−15), then substitute into x2−(sum)x+(product)=0 to instantly get x2+2x−15=0.
Students often make a sign error while writing the coefficient of the middle term by forgetting that the formula uses negative of the sum of roots: x2−(α+β)x.
Identify the given roots
Let the two roots of the quadratic equation be given as α=3 and β=−5.
α=3,β=−5
Calculate the sum and product of roots
Find the sum of the roots α+β=3+(−5)=−2 and the product of the roots αβ=3×(−5)=−15.
α+β=−2,αβ=−15
Apply the standard quadratic equation formula
Substitute the sum and product into the standard equation formula: x2−(α+β)x+αβ=0.
x2−(−2)x+(−15)=0
Simplify the equation
Simplify the signs to arrive at the final quadratic equation: x2+2x−15=0.
x2+2x−15=0
A is correct because the quadratic equation formed using the roots 3 and -5 is x2+2x−15=0.
Understanding the relationship between roots and coefficients is crucial for higher-degree polynomials and finding maximum or minimum values of quadratic expressions in CBT-1 exams.
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