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y = x2 + bx + 256 cuts the x axis at (h, 0) and (k, 0). If h and k are integers, what is the least value of b?
- a)-32
- b)-256
- c)-255
- d)-257
- e)0
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
y = x2+ bx + 256 cuts the x axis at (h, 0) and (k, 0). If h and k are ...
Step 1: Understand the equation and the points (h, 0) and (k, 0)
The given equation is a quadratic equation. A quadratic equation when plotted on a graph sheet (x - y plane) will result in a parabola.
The roots of the quadratic equation are computed by equating the quadratic expression to 0. i.e., the roots are the values that 'x' take when y = 0
So, the roots of the quadratic equation are the points where the parabola cuts the x-axis.
The question mentions that the curve described by the equation cuts the x-axis at (h, 0) and (k, 0). So, h and k are the roots of the quadratic equation.
For quadratic equations of the form ax2 + bx + c = 0, the sum of the roots =
The sum of the roots of this equation is
Note : Higher the value of 'b', i.e., higher the sum of the roots of this quadratic equation, lower the value of b.
The roots of the quadratic equation are computed by equating the quadratic expression to 0. i.e., the roots are the values that 'x' take when y = 0
So, the roots of the quadratic equation are the points where the parabola cuts the x-axis.
The question mentions that the curve described by the equation cuts the x-axis at (h, 0) and (k, 0). So, h and k are the roots of the quadratic equation.
For quadratic equations of the form ax2 + bx + c = 0, the sum of the roots =
The sum of the roots of this equation is
Note : Higher the value of 'b', i.e., higher the sum of the roots of this quadratic equation, lower the value of b.
For quadratic equations of the form ax2 + bx + c = 0, the product of roots = c/a.
Therefore, the product of the roots of this equation = 256/1 = 256.
i.e., h × k = 256 h and k are both integers.
So, h and k are both integral factors of 256.
Therefore, the product of the roots of this equation = 256/1 = 256.
i.e., h × k = 256 h and k are both integers.
So, h and k are both integral factors of 256.
Step 2: List possible values of h and k and find the least value of ‘b’
This is the step in which number properties concepts kick in. 256 can be expressed as product of two numbers in the following ways:
1 × 256
2 × 128
4 × 64
8 × 32
16 × 16
1 × 256
2 × 128
4 × 64
8 × 32
16 × 16
The sum of the roots is maximum when the roots are 1 and 256 and the maximum sum is 1 + 256 = 257.
∴ The least value possible for b is -257.
∴ The least value possible for b is -257.
Choice D is the correct answer.
Most Upvoted Answer
y = x2+ bx + 256 cuts the x axis at (h, 0) and (k, 0). If h and k are ...
Understanding the Problem
The quadratic equation given is:
y = x² + bx + 256
This equation intersects the x-axis at points (h, 0) and (k, 0). Here, h and k are the roots of the equation.
Using Vieta's Formulas
According to Vieta’s formulas:
- The sum of the roots (h + k) = -b
- The product of the roots (h * k) = 256
Finding Integer Roots
We need to explore integer pairs (h, k) that satisfy the product condition:
h * k = 256
The integer factor pairs of 256 are:
- (1, 256)
- (2, 128)
- (4, 64)
- (8, 32)
- (16, 16)
- (-1, -256)
- (-2, -128)
- (-4, -64)
- (-8, -32)
- (-16, -16)
Calculating b for Each Pair
For each pair (h, k), we can calculate b using the formula:
b = -(h + k)
Now, let's evaluate the pairs:
- (1, 256) → b = -(1 + 256) = -257
- (2, 128) → b = -(2 + 128) = -130
- (4, 64) → b = -(4 + 64) = -68
- (8, 32) → b = -(8 + 32) = -40
- (16, 16) → b = -(16 + 16) = -32
- (-1, -256) → b = -(-1 - 256) = 257
- (-2, -128) → b = -(-2 - 128) = 130
- (-4, -64) → b = -(-4 - 64) = 68
- (-8, -32) → b = -(-8 - 32) = 40
- (-16, -16) → b = -(-16 - 16) = 32
Finding the Least Value of b
From these calculations, the least value of b occurs with the pair (1, 256) or (2, 128).
The minimum b value is:
-257
Thus, the correct answer is option 'D': -257.
The quadratic equation given is:
y = x² + bx + 256
This equation intersects the x-axis at points (h, 0) and (k, 0). Here, h and k are the roots of the equation.
Using Vieta's Formulas
According to Vieta’s formulas:
- The sum of the roots (h + k) = -b
- The product of the roots (h * k) = 256
Finding Integer Roots
We need to explore integer pairs (h, k) that satisfy the product condition:
h * k = 256
The integer factor pairs of 256 are:
- (1, 256)
- (2, 128)
- (4, 64)
- (8, 32)
- (16, 16)
- (-1, -256)
- (-2, -128)
- (-4, -64)
- (-8, -32)
- (-16, -16)
Calculating b for Each Pair
For each pair (h, k), we can calculate b using the formula:
b = -(h + k)
Now, let's evaluate the pairs:
- (1, 256) → b = -(1 + 256) = -257
- (2, 128) → b = -(2 + 128) = -130
- (4, 64) → b = -(4 + 64) = -68
- (8, 32) → b = -(8 + 32) = -40
- (16, 16) → b = -(16 + 16) = -32
- (-1, -256) → b = -(-1 - 256) = 257
- (-2, -128) → b = -(-2 - 128) = 130
- (-4, -64) → b = -(-4 - 64) = 68
- (-8, -32) → b = -(-8 - 32) = 40
- (-16, -16) → b = -(-16 - 16) = 32
Finding the Least Value of b
From these calculations, the least value of b occurs with the pair (1, 256) or (2, 128).
The minimum b value is:
-257
Thus, the correct answer is option 'D': -257.
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