Question about solving a quadratic a word problem

Question

The question is the sum of two positive number is $21$. Fifteen less than twice the square of the smaller number gives the larger number. If $x$ represents the smaller number, write a system of equations to model this information and solve the system to determine the two positive numbers.

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Here's my solution to the problem:

Let the smaller and larger number be $x$.

$$15 - 2 \sqrt x = x $$ $$x = 21$$ Then how do I solve it? Because I won't be able solve it then.

Answer 1

Notice, $x$ is the smaller number let the larger number be $y$.

Then we have following conditions:

$1.)$ Sum of numbers is $21$ hence $$x+y=21\tag 1$$

$2.)$
$$2x^2-15=y\tag 2$$ Now, substituting the value $y=21-x$ from $(1)$ into $(2)$, we get $$2x^2-15=21-x$$ $$2x^2+x-36=0$$ $$2x^2+9x-8x-36=0$$ $$x(2x+9)-4(2x+9)=0$$ $$(2x+9)(x-4)=0$$ $$x=4, \ \frac{-9}{2} $$ But the numbers are positive hence we get $x=\color{red}{4}$

Then the corresponding values of $y$ $$y=21-x=21-4=\color{red}{17}$$

Answer 2

We have $x+y=21$ and $2x^2-15=y$. Substitute the latter into the former, and we get $x+(2x^2-15)=21,$ or $2x^2+x-36=0.$ Can you work from there to solve for $x$?

Answer 3

"Fifteen less than twice the square of the smaller number gives the larger number."

Key tips on translations:

• Two different numbers need two different variables (not $x$ for both).
• "Square" ($x^2$) is not the same as "square root" ($\sqrt{x}$).
• "Less than" reverses the order of the terms mentioned.

Therefore this second sentence is translated as: $2x^2 - 15 = y$, and the first sentence means $x + y = 21$. You should find this easier to solve. Recommend that you practice some translations from English to math, for example here: https://www.khanacademy.org/math/algebra/introduction-to-algebra/writing-expressions-tutorial/e/writing_expressions_1