Question

The function f(x) = 2x2 + 3x + 5, when evaluated, gives a value of 19. What is the function’s input value?

Answer 1

`x=2, \quad x=-(7)/(2)`

Explanation:

Set the function to 19:

`\implies 2x^2+3x+5=19`

Subtract 19 from both sides:

`\implies 2x^2+3x-14=0`

To factor a quadratic in the form
`ax^2+bx+c`

Find two numbers that multiply to
`ac` and sum to
`b`: 7 and -4

Rewrite
`b` as the sum of these two numbers:

`\implies 2x^2-4x+7x-14=0`

Factorize the first two terms and the last two terms separately:

`\implies 2x(x-2)+7(x-2)=0`

Factor out the common term (x - 2):

`\implies (2x+7)(x-2)=0`

Therefore the function's input values that when evaluated give a value of 19 are:

`(2x+7)=0 \implies x=-(7)/(2)`

`(x-2)=0 \implies x=2`

Answer 2

-7/2 or 2 are inputs that give 19 as output.

Explanation:

The problem gives us a quadratic function
`\displaystyle \large{f(x)=2x^2+3x+5}`. When its output is 19, we want to know the input value(s).

Since an output which is f(x) = 19. Therefore:

`\displaystyle \large{19=2x^2+3x+5}`

Rearrange the expression in quadratic equation.

`\displaystyle \large{0=2x^2+3x+5-19}\\\\\displaystyle \large{0=2x^2+3x-14}\\\\\displaystyle \large{2x^2+3x-14=0}`

Factor the expression.

`\displaystyle \large{(2x+7)(x-2)=0}`

Solve like linear equation which we get:

`\displaystyle \large{x=-(7)/(2), 2}`

If you input these x-values in the function, you will get 19 as the output which satisfies the condition.

Hence, inputs are -7/2, 2