Question

Find the value of k for which the curve 2x*-3x=K meets the x-axis at one point only

Answer 1

a. The value of
`\( k \)` is -27.

b. The value of
`\( k \)` is
`\( (9)/(8) \).`

How did we get these values?

Let's first address part (a) where we need the curve to pass through the point (4, -7).

For a point (x, y) on the curve, the x and y coordinates must satisfy the equation of the curve. So, for the given point (4, -7), we can substitute x = 4 and y = -7 into the equation y =
`2x^2 - 3x + k:`

`\[ -7 = 2(4)^2 - 3(4) + k \]`

Solving this equation will give us the value of k. Let's calculate:

`\[ -7 = 32 - 12 + k \]`

Combine like terms:

`\[ -7 = 20 + k \]`

Subtract 20 from both sides:

`\[ k = -27 \]`

So, for part (a), the value of
`\( k \)` is -27.

Now, let's move on to part (b) where the curve meets the x-axis at one point only. This implies that the discriminant of the quadratic equation
`\(2x^2 - 3x + k = 0\)` should be zero. The discriminant
`(\( \Delta \))` of a quadratic equation
`\(ax^2 + bx + c = 0\)` is given by
`\( \Delta = b^2 - 4ac \).`

For our equation
`\(2x^2 - 3x + k = 0\)`, the discriminant is:

`\[ \Delta = (-3)^2 - 4(2)(k) \]`

We want this to be zero:

`\[ 9 - 8k = 0 \]`

Solve for
`\( k \)`:

`\[ 8k = 9 \]`

`\[ k = (9)/(8) \]`

So, for part (b), the value of
`\( k \)` is
`\( (9)/(8) \).`

Complete question:

Find the value of k for which the curve y=2x^2-3x+k (a) passes through the point (4,-7) (b) meets the x-axis at one point only.