The solutions to the quadratic equation x² + 6x = 40 is given by: Option D: x = −4 and x = 10
Solution of an equation is typically those values of its variables, for which the considered equation is true.
How to find the factors of a quadratic expression?
If the given quadratic expression is of the form
[tex]ax^2 + bx + c = 0[/tex]
then its factored form is obtained by two numbers alpha( α ) and beta( β) such that:
[tex]b = \alpha + \beta \\ ac =\alpha \times \beta[/tex]
Then writing b in terms of alpha and beta would help us getting common factors out.
It is not always possible to split 'b' in such terms. That's where we can use the method of completing the squares.
For this case, the quadratic equation in consideration is:
x² + 6x = 40
Rewriting it gives:
[tex]x^2 + 6x = 40\\x^2 + 6x -40 = 0\\[/tex]
Now, 6 needs to be written as sum of two terms whose product is -40
Negative result shows we need to use subtraction.
40 is writable in product as product of 1 and 40, of 2 and 20, of 4 and 10 (yes, its difference is 6). 10 + (-4) = 6 and 10 times -4 = -40
Thus, we get;
[tex]x^2 + 6x - 40 = 0\\\\x^2 + 10x - 4x - 40 = 0\\\\x(x+10) -4(x+10) =0\\\\(x-4)(x+10) = 0\\\\(x-4 = 0, x + 10 = 0\\\\x = 4, x = -10[/tex]
Thus, the solutions to the quadratic equation x² + 6x = 40 is given by: Option D: x = −4 and x = 10
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