Which one of the following equations does not have real roots ?
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Step-by-Step Solution
Step 1: Understand the condition for real roots
A quadratic equation of the form ax2+bx+c=0 has real roots if its discriminant, denoted by D, is greater than or equal to zero. The discriminant is calculated using the formula D=b2−4ac. If D<0, the roots are not real; they are complex.
Step 2: Analyze the first equation
For the first equation, x2−4x+32=0, we identify a=1, b=−4, and c=32. We then calculate the discriminant D. Since 122≈12×1.414=16.968, D=16−16.968=−0.968, which is less than 0. Therefore, this equation does not have real roots.
Step 3: Analyze the second equation
For the second equation, x2+4x−32=0, we have a=1, b=4, and c=−32. Calculating the discriminant, we get D=16+122. Since 122 is a positive number, D will be greater than 0, meaning this equation has real roots.
Step 4: Analyze the third equation
For the third equation, x2−4x−32=0, we have a=1, b=−4, and c=−32. The discriminant is D=16+122. This is also a positive value, indicating that this equation has real roots.
Step 5: Analyze the fourth equation
For the fourth equation, 3x2+43x+4=0, we have a=3, b=43, and c=4. The discriminant is D=(43)2−4(3)(4)=48−48=0. Since D=0, this equation has real and equal roots.