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Advanced Math / Nonlinear equations in one variable and systems of equations in two variables Difficulty: Hard

In the xy-plane, a line with equation $2 y equals c$ for some constant $c$ intersects a parabola at exactly one point. If the parabola has equation $y = - 2 x^{2} + 9 x$ , what is the value of $c$ ?

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Explanation

The correct answer is $StartFraction 81 Over 4 EndFraction$ . The given linear equation is $2 y equals c$ . Dividing both sides of this equation by $2$ yields $y = \frac{c}{2}$ . Substituting $StartFraction c Over 2 EndFraction$ for $y$ in the equation of the parabola yields $\frac{c}{2} = - 2 x^{2} + 9 x$ . Adding $2 x squared$ and $minus 9 x$ to both sides of this equation yields $2 x^{2} - 9 x + \frac{c}{2} = 0$ . Since it’s given that the line and the parabola intersect at exactly one point, the equation $2 x^{2} - 9 x + \frac{c}{2} = 0$ must have exactly one solution. An equation of the form $A x^{2} + B x + C = 0$ , where $A$ , $B$ , and $C$ are constants, has exactly one solution when the discriminant, $upper B squared minus 4 upper A upper C$ , is equal to $0$ . In the equation $2 x^{2} - 9 x + \frac{c}{2} = 0$ , where $upper A equals 2$ , $B = - 9$ , and $C = \frac{c}{2}$ , the discriminant is ${(- 9)}^{2} - 4 (2) (\frac{c}{2})$ . Setting the discriminant equal to $0$ yields ${(- 9)}^{2} - 4 (2) (\frac{c}{2}) = 0$ , or $81 - 4 c = 0$ . Adding $4 c$ to both sides of this equation yields $81 equals 4 c$ . Dividing both sides of this equation by $4$ yields $c = \frac{81}{4}$ . Note that 81/4 and 20.25 are examples of ways to enter a correct answer.