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Question Find the least possible value of integer constant kk in the equation x(kx−56)=−16x(k x-56)=-16 with no real solution.

Studdy Solution

STEP 1

Assumptions
1. The given equation is x(kx−56)=−16 x(kx - 56) = -16 .
2. k k is an integer constant.
3. The equation has no real solution.

STEP 2

We will first expand the equation to its standard quadratic form ax2+bx+c=0 ax^2 + bx + c = 0 .
x(kx−56)=−16 x(kx - 56) = -16

STEP 3

Distribute x x across the parentheses.
x⋅kx−x⋅56=−16 x \cdot kx - x \cdot 56 = -16

STEP 4

Combine like terms to get the quadratic equation.
kx2−56x+16=0 kx^2 - 56x + 16 = 0

STEP 5

For a quadratic equation ax2+bx+c=0 ax^2 + bx + c = 0 , it has no real solutions if the discriminant b2−4ac b^2 - 4ac is less than 0.

STEP 6

Identify the coefficients a a , b b , and c c from the quadratic equation.
a=k,b=−56,c=16 a = k, \quad b = -56, \quad c = 16

STEP 7

Write down the discriminant formula with the identified coefficients.
Δ=b2−4ac \Delta = b^2 - 4ac

STEP 8

Substitute the values of a a , b b , and c c into the discriminant formula.
Δ=(−56)2−4⋅k⋅16 \Delta = (-56)^2 - 4 \cdot k \cdot 16

STEP 9

Calculate the square of −56 -56 .
(−56)2=3136 (-56)^2 = 3136

STEP 10

Substitute the calculated square into the discriminant formula.
Δ=3136−4⋅k⋅16 \Delta = 3136 - 4 \cdot k \cdot 16

STEP 11

Simplify the expression by multiplying 4 4 and 16 16 .
Δ=3136−64k \Delta = 3136 - 64k

STEP 12

For the equation to have no real solutions, the discriminant must be less than 0.
Δ<0 \Delta < 0

STEP 13

Substitute the expression for the discriminant into the inequality.
3136−64k<0 3136 - 64k < 0

STEP 14

Solve the inequality for k k .
−64k<−3136 -64k < -3136

STEP 15

Divide both sides of the inequality by −64 -64 to isolate k k , remembering to reverse the inequality sign because we are dividing by a negative number.
k>−3136−64 k > \frac{-3136}{-64}

STEP 16

Calculate the division to find the least possible value of k k .
k>−3136−64 k > \frac{-3136}{-64}

STEP 17

Perform the division.
k>−3136−64=49 k > \frac{-3136}{-64} = 49

STEP 18

Since k k must be an integer and greater than 49, the least possible integer value of k k is 50.
The least possible value of k k is 50.

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