# What is the value of a if the graph of y = ax^2 + 2x + 4 intercepts the line y + x = 6 at only one point.

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The graph of the curve y = ax^2 + 2x + 4 and the line y + x = 6 intersect at only one point.

From y + x = 6, the value of the dependent variable is y = 6 - x.

Equating this to y = ax^2 + 2x + 4 gives:

6 - x = ax^2 + 2x + 4

=> ax^2 + 3x - 2= 0

Any quadratic equation ax^2 + bx + c = 0 has two equal roots if `b^2 - 4*a*c = 0` . Substitute the values for a, b and c from ax^2 + 3x - 2 = 0.

`3^2 + 4*a*2 = 0 => a = -9/8`

The curve y = -9x^2/8 + 2x + 4 intersects the line y + x = 6 once.

**The required value of a = -9/8**