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If one of the roots of the quadratic equation x2 + mx + 24 = 0 is 1.5, then what is the value of m?
- a)-22.5
- b)16
- c)-10.5
- d)-17.5
- e)Cannot be determined
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
If one of the roots of the quadratic equation x2+ mx + 24 = 0 is 1.5, ...
Approach to solve: If 1.5 is a root of the quadratic equation, substituting x = 1.5 in the equation will satisfy the equation.
The given quadratic equation is x2 + mx + 24 = 0
Substitute x = 1.5 in the above equation because 1.5 is a root of the equation.
(1.5)2 + 1.5m + 24 = 0
2.25 + 1.5m + 24 = 0
1.5m = -26.25 Or m = −26.251.5−26.251.5 = -17.5
Substitute x = 1.5 in the above equation because 1.5 is a root of the equation.
(1.5)2 + 1.5m + 24 = 0
2.25 + 1.5m + 24 = 0
1.5m = -26.25 Or m = −26.251.5−26.251.5 = -17.5
Alternative Method
Step 1: Sum and Product of Roots of Quadratic Equations Theory
For quadratic equations of the form ax2 + bx + c = 0, whose roots are α and β,
Sum of the roots, α + β =
, and product of the roots, αβ = c/a.
From the question stem, we know that one of the roots is 1.5. Let α be 1.5.
Sum of the roots, α + β =
, and product of the roots, αβ = c/a.From the question stem, we know that one of the roots is 1.5. Let α be 1.5.
Step 2: Compute the second root of the equation
Product of the roots of the quadratic equation x2 + mx + 24 = 0 is c/a = (24/1) = 24.
i.e., α * β = 24 where α is 1.5.
1.5 * β = 24
β = (24/1.5)
β = 16
i.e., α * β = 24 where α is 1.5.
1.5 * β = 24
β = (24/1.5)
β = 16
Step 3: Compute the value of ‘m’
In the given equation, m is the co-efficient of the x term.
We know that the sum of the roots of quadratic equations of the form ax2 + bx + c = 0 is −b/a = −m/1 = -m
Sum of the roots = 16 + 1.5 = 17.5
Sum of the roots = -m
If –m = 17.5, the value of m = -17.5
We know that the sum of the roots of quadratic equations of the form ax2 + bx + c = 0 is −b/a = −m/1 = -m
Sum of the roots = 16 + 1.5 = 17.5
Sum of the roots = -m
If –m = 17.5, the value of m = -17.5
Choice D is the correct answer.
Most Upvoted Answer
If one of the roots of the quadratic equation x2+ mx + 24 = 0 is 1.5, ...
Understanding the Quadratic Equation
A quadratic equation can be expressed in the standard form as:
- x^2 + mx + c = 0
In this case, the equation is:
- x^2 + mx + 24 = 0
Where c = 24.
Given Information
- One of the roots of the equation is 1.5.
Using Vieta's Formulas
According to Vieta's formulas, for a quadratic equation of the form x^2 + bx + c = 0:
- The sum of the roots (r1 + r2) = -b
- The product of the roots (r1 * r2) = c
Here, one root (r1) is 1.5, and we need to find the other root (r2).
Calculating the Product of Roots
Since the product of the roots (r1 * r2) equals 24:
- 1.5 * r2 = 24
To find r2, divide both sides by 1.5:
- r2 = 24 / 1.5
- r2 = 16
Finding the Value of m
Now, we can find the sum of the roots:
- r1 + r2 = 1.5 + 16 = 17.5
According to Vieta's formulas:
- -m = 17.5
- Therefore, m = -17.5
Conclusion
The value of m in the quadratic equation x^2 + mx + 24 = 0, given that one root is 1.5, is:
- m = -17.5
Thus, the correct answer is:
- Option D: -17.5
A quadratic equation can be expressed in the standard form as:
- x^2 + mx + c = 0
In this case, the equation is:
- x^2 + mx + 24 = 0
Where c = 24.
Given Information
- One of the roots of the equation is 1.5.
Using Vieta's Formulas
According to Vieta's formulas, for a quadratic equation of the form x^2 + bx + c = 0:
- The sum of the roots (r1 + r2) = -b
- The product of the roots (r1 * r2) = c
Here, one root (r1) is 1.5, and we need to find the other root (r2).
Calculating the Product of Roots
Since the product of the roots (r1 * r2) equals 24:
- 1.5 * r2 = 24
To find r2, divide both sides by 1.5:
- r2 = 24 / 1.5
- r2 = 16
Finding the Value of m
Now, we can find the sum of the roots:
- r1 + r2 = 1.5 + 16 = 17.5
According to Vieta's formulas:
- -m = 17.5
- Therefore, m = -17.5
Conclusion
The value of m in the quadratic equation x^2 + mx + 24 = 0, given that one root is 1.5, is:
- m = -17.5
Thus, the correct answer is:
- Option D: -17.5
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