Find the range of values for $p$ such that $x^2 - 2x + 3 - p = 0$ has no real roots. - Scottish Highers Maths - Question 2 - 2016
Question 2
Find the range of values for $p$ such that $x^2 - 2x + 3 - p = 0$ has no real roots.
Worked Solution & Example Answer:Find the range of values for $p$ such that $x^2 - 2x + 3 - p = 0$ has no real roots. - Scottish Highers Maths - Question 2 - 2016
Step 1
Step 1: Use the Discriminant
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Answer
To determine the conditions for the quadratic equation x2−2x+(3−p)=0 to have no real roots, we start by using the discriminant (extD). The discriminant is given by:
D=b2−4ac
Where a=1, b=−2, and c=3−p.
Step 2
Step 2: Simplify the Discriminant
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Answer
Calculating the discriminant, we have:
D=(−2)2−4(1)(3−p)
This simplifies to:
D=4−12+4p=4p−8
Step 3
Step 3: Apply the Condition for No Real Roots
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Answer
For the quadratic to have no real roots, we require that the discriminant is less than zero:
4p−8<0
Solving this inequality leads to:
4p<8p<2
Step 4
Step 4: State the Range of Values
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Answer
Thus, the range of values for p such that the equation has no real roots is:
p<2
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