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Answer:
-7/2 or 2 are inputs that give 19 as output.
Step-by-step explanation:
The problem gives us a quadratic function [tex]\displaystyle \large{f(x)=2x^2+3x+5}[/tex]. When its output is 19, we want to know the input value(s).
Since an output which is f(x) = 19. Therefore:
[tex]\displaystyle \large{19=2x^2+3x+5}[/tex]
Rearrange the expression in quadratic equation.
[tex]\displaystyle \large{0=2x^2+3x+5-19}\\\\\displaystyle \large{0=2x^2+3x-14}\\\\\displaystyle \large{2x^2+3x-14=0}[/tex]
Factor the expression.
[tex]\displaystyle \large{(2x+7)(x-2)=0}[/tex]
Solve like linear equation which we get:
[tex]\displaystyle \large{x=-\dfrac{7}{2}, 2}[/tex]
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
Answer:
[tex]x=2, \quad x=-\dfrac{7}{2}[/tex]
Step-by-step explanation:
Set the function to 19:
[tex]\implies 2x^2+3x+5=19[/tex]
Subtract 19 from both sides:
[tex]\implies 2x^2+3x-14=0[/tex]
To factor a quadratic in the form [tex]ax^2+bx+c[/tex]
Find two numbers that multiply to [tex]ac[/tex] and sum to [tex]b[/tex]: 7 and -4
Rewrite [tex]b[/tex] as the sum of these two numbers:
[tex]\implies 2x^2-4x+7x-14=0[/tex]
Factorize the first two terms and the last two terms separately:
[tex]\implies 2x(x-2)+7(x-2)=0[/tex]
Factor out the common term (x - 2):
[tex]\implies (2x+7)(x-2)=0[/tex]
Therefore the function's input values that when evaluated give a value of 19 are:
[tex](2x+7)=0 \implies x=-\dfrac{7}{2}[/tex]
[tex](x-2)=0 \implies x=2[/tex]