. x2 – 2x + 4 = 0 discriminant: 2. 4×2 – 5x + 12 = 0 5x + 12 = 0 discriminant: 3. x2 + 4x – 21 = 0 discriminant: 4. x2 – 9x + 7 = 0 discriminant: 5. x2 + 3x + 3 = 0 discriminant:
The Correct Answer and Explanation is:
Let’s go through each quadratic equation and compute the discriminant using the formula:Discriminant=D=b2−4ac\text{Discriminant} = D = b^2 – 4acDiscriminant=D=b2−4ac
This value helps determine the nature of the roots of a quadratic equation ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0:
- If D>0D > 0D>0: Two real and distinct roots
- If D=0D = 0D=0: One real and repeated (equal) root
- If D<0D < 0D<0: Two complex (non-real) conjugate roots
1. x2−2x+4=0x^2 – 2x + 4 = 0x2−2x+4=0
Here, a=1,b=−2,c=4a = 1, b = -2, c = 4a=1,b=−2,c=4D=(−2)2−4(1)(4)=4−16=−12D = (-2)^2 – 4(1)(4) = 4 – 16 = -12D=(−2)2−4(1)(4)=4−16=−12
✅ Discriminant = -12 (Complex roots)
2. 4×2−5x+12=04x^2 – 5x + 12 = 04×2−5x+12=0
Here, a=4,b=−5,c=12a = 4, b = -5, c = 12a=4,b=−5,c=12D=(−5)2−4(4)(12)=25−192=−167D = (-5)^2 – 4(4)(12) = 25 – 192 = -167D=(−5)2−4(4)(12)=25−192=−167
✅ Discriminant = -167 (Complex roots)
Note: The line “5x + 12 = 0” seems unrelated; it’s not part of this quadratic equation.
3. x2+4x−21=0x^2 + 4x – 21 = 0x2+4x−21=0
Here, a=1,b=4,c=−21a = 1, b = 4, c = -21a=1,b=4,c=−21D=(4)2−4(1)(−21)=16+84=100D = (4)^2 – 4(1)(-21) = 16 + 84 = 100D=(4)2−4(1)(−21)=16+84=100
✅ Discriminant = 100 (Real and distinct roots)
4. x2−9x+7=0x^2 – 9x + 7 = 0x2−9x+7=0
Here, a=1,b=−9,c=7a = 1, b = -9, c = 7a=1,b=−9,c=7D=(−9)2−4(1)(7)=81−28=53D = (-9)^2 – 4(1)(7) = 81 – 28 = 53D=(−9)2−4(1)(7)=81−28=53
✅ Discriminant = 53 (Real and distinct roots)
5. x2+3x+3=0x^2 + 3x + 3 = 0x2+3x+3=0
Here, a=1,b=3,c=3a = 1, b = 3, c = 3a=1,b=3,c=3D=(3)2−4(1)(3)=9−12=−3D = (3)^2 – 4(1)(3) = 9 – 12 = -3D=(3)2−4(1)(3)=9−12=−3
✅ Discriminant = -3 (Complex roots)
Explanation
The discriminant of a quadratic equation is a powerful tool that determines the type of solutions the equation has. Given the general form ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0, the discriminant is calculated using the formula D=b2−4acD = b^2 – 4acD=b2−4ac.
This value directly influences the nature of the roots:
- If D>0D > 0D>0, the roots are real and distinct, meaning the graph of the parabola intersects the x-axis at two distinct points.
- If D=0D = 0D=0, the roots are real and equal, indicating the parabola just touches the x-axis at a single point (called the vertex).
- If D<0D < 0D<0, the roots are complex (non-real), and the parabola does not intersect the x-axis at all.
In the examples above:
- Equations 1, 2, and 5 all have negative discriminants, implying complex roots. These equations do not produce real-number solutions and their graphs remain above or below the x-axis without touching it.
- Equations 3 and 4 have positive discriminants (100 and 53, respectively), meaning they each have two real and distinct roots. These equations produce real-number solutions, which is evident in the form of two x-intercepts on the graph.
Understanding the discriminant not only helps predict the number and type of solutions without solving the equation but also aids in sketching the graph and analyzing the behavior of quadratic functions in algebra and calculus.
