Difference between revisions of "2005 AMC 10A Problems/Problem 10"

Revision as of 17:04, 11 April 2019

Problem

There are two values of $a$ for which the equation $4x^2 + ax + 8x + 9 = 0$ has only one solution for $x$ . What is the sum of those values of $a$ ?

$\mathrm{(A) \ } -16\qquad \mathrm{(B) \ } -8\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } 8\qquad \mathrm{(E) \ } 20$

Solution

A quadratic equation has exactly one root if and only if it is a perfect square. So set

$4x^2 + ax + 8x + 9 = (mx + n)^2$

$4x^2 + ax + 8x + 9 = m^2x^2 + 2mnx + n^2$

Two polynomials are equal only if their coefficients are equal, so we must have

$m^2 = 4, n^2 = 9$

$m = \pm 2, n = \pm 3$

$a + 8= 2mn = \pm 2\cdot 2\cdot 3 = \pm 12$

$a = 4$ or $a = -20$ .

So the desired sum is $(4)+(-20)=-16 \Longrightarrow \mathrm{(A)}$

Alternatively, note that whatever the two values of $a$ are, they must lead to equations of the form $px^2 + qx + r =0$ and $px^2 - qx + r = 0$ . So the two choices of $a$ must make $a_1 + 8 = q$ and $a_2 + 8 = -q$ so $a_1 + a_2 + 16 = 0$ and $a_1 + a_2 = -16\Longrightarrow \mathrm{(A)}$ .

Alternate Solution

Since this quadratic must have a double root, the discriminant of the quadratic formula for this quadratic must be 0. Therefore, we must have $(a+8)^2 - 4(4)(9) = 0 \implies a^2 + 16a - 80.$ We can use the quadratic formula to solve for its roots (we can ignore the things in the radical sign as they will cancel out due to the $\pm$ sign when added). So we must have ${-16 + \sqrt{\text{something}}}{2} + \frac{-16 - \sqrt{\text{something}}}{2}.$ Therefore, we have \implies \boxed{A}.$$ (Error compiling LaTeX. Unknown error_msg)

Solution 3

There is only one positive value for k such that the quadratic equation would have only one solution. k-8 and -k-8 are the values of a. I am certain we al know what -8-8 is. That’s correct, \implies \boxed{A}.$

@@ Line 32: / Line 32: @@
 <cmath> (a+8)^2 - 4(4)(9) = 0 \implies a^2 + 16a - 80. </cmath> We can use the quadratic formula to solve for its roots (we can ignore the things in the radical sign as they will cancel out due to the <math>\pm</math> sign when added). So we must have
 <cmath> {-16 + \sqrt{\text{something}}}{2} + \frac{-16 - \sqrt{\text{something}}}{2}. </cmath>
-Therefore, we have \implies \boxed{A}.<math>
+Therefore, we have \implies \boxed{A}.<math></math>
 == Solution 3==
 There is only one positive value for k such that the quadratic equation would have only one solution.
-k-8 and -k-8 are the values of a. I am certain we al know what -8-8 is. That’s correct, \implies \boxed{A}.</math>
+k-8 and -k-8 are the values of a. I am certain we al know what -8-8 is. That’s correct, \implies \boxed{A}.$
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