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What is the largest number c such that 2x^2 + 5x + c = x^2 - 4x has at least  one real solution?  Express your answer as a common fraction.

 Jan 10, 2023
 #1
avatar+34 
+2

Hmmm...

 

We have the equation:

$$2x^2 + 5x + c = x^2 - 4x$$

We subtract x^2 from both sides:

$$x^2 + 5x + c = -4x$$
We add 4x to both sides:

$$x^2 + 9x + c = 0$$

 

Now, I want to clarify one thing, when finding the largest product, there is a pretty easy way to do this with intuition.

Try multiplying:

 

$$7*\cdot3 = 21$$

$$6\cdot4 = 24$$

$$5\cdot5 = 25$$

 

When doing this similar multiplication process with other numbers, you will realize that the closer the numbers are, the greater their product is:

 

So, when we want the max value of our constant, we have to use the closest numbers together.

We also notice that our two numbers (r and s) have to equal 9. Since we need the closest numbers, then r = s.

$$2r = 9$$

$$r = 4.5$$

So our equation would be in the form:

$$(x + 4.5)(x + 4.5) = x^2 + 9x + 20.25$$

 

Therefore the greatest constant value is 20.25.

Expressing this answer in a fraction can either be 20 1/4 or 81/4

Please tell me if I am incorrect as I would be happy to fix it!

 Jan 10, 2023
 #2
avatar+2526 
+2

Here's another way: 

 

The equation simplifies to \(x^2 + 9x + c = 0\)

 

The equation has at least 1 real root when the discriminant \(b^2 - 4ac \geq 0\)

 

Substituting in what we have gives us \(81 - 4c \geq 0 \), meaning \(c \leq 20.25\), so the maximum value of c is \(20.25 = \color{brown}\boxed{81\over 4}\)

 Jan 10, 2023
 #3
avatar+34 
+1

Oh, right, you could also solve it using discriminants. Didn't even think of that cheeky

Hmmmm  Jan 10, 2023

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